Generalizations and refinements of three-tuple Diamond-Alpha integral Hölder’s inequality on time scales

Abstract

In this paper, based on the existing Hölder’s inequality, some new three-tuple diamond-alpha integral Hölder’s inequalities on time scales are proposed and the related theorems and corollaries are given. At the same time, we also give the relevant conclusions and proof of n-tuple diamond-alpha integral Hölder’s inequalities on time scales.

1 Introduction

Let $f(\delta )>0, g(\delta )>0, p>1, 1/p+1/q=1$. If $f(\delta )$ and $g(\delta )$ are continuous real-valued functions on $[\xi , \sigma ]$, then

$$ \int _{\xi }^{\sigma } f(\delta )g(\delta ) \,d\delta \leq \biggl( \int _{\xi }^{\sigma } f^{p}(x)\,d\delta \biggr)^{1/p} \biggl( \int _{\xi }^{ \sigma }g^{q}(x)\,d\delta \biggr)^{1/q}. $$

This famous Hölder inequality is extended in article [1] to the diamond-α integral Hölder inequality on time scales, in the following form:

Let $f,g,h:[\xi , \sigma ]\to \mathbb{R}$ be $\Diamond _{\alpha }$-integrable functions, and $1/p+1/q=1$ with $p>1$, then

$$ \int _{\xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert f(\delta )g(\delta ) \bigr\vert \Diamond _{ \alpha } \delta \leq \biggl( \int _{\xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert f(\delta ) \bigr\vert ^{p}\Diamond _{\alpha } \delta \biggr)^{\frac{1}{p}} \biggl( \int _{ \xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert g(\delta ) \bigr\vert ^{q}\Diamond _{\alpha } \delta \biggr)^{\frac{1}{q}}. $$

Since Hilger [2] proposed the time-scale theory in 1998, many researchers [3, 4] have made extensive promotions and applications of his theory. The classical analytic inequality [5–8], especially Hölder’s inequality, plays a very important role in modern mathematics. Due to the importance of both, more and more scholars [9–15] have studied the intersections of two inequalities. The purpose of this article is to derive some generalizations and refinements of the three-tuple diamond-α integral Hölder inequality on time scales. The relevant conclusions of the n-tuple diamond-α integral Hölder inequality on time scales are also given.

2 Main lemmas

Before the main results are given in this paper, we need to introduce the following lemmas, which are helpful for the results of this paper.

Lemma 2.1

([16])

Let $\sum_{j=1}^{m}\frac{1}{p_{j}}=1, \lambda _{j} \geq 0\ (j=1,2,\ldots ,m)$. Then

(1) for $p_{j}>1$, we have

$$ \prod_{j=1}^{m}\lambda _{j}\leq \sum_{j=1}^{m} \frac{\lambda _{j}^{p_{j}}}{p _{j}}, $$

(1)

(2) for $0< p_{m}<1, p_{j}<0\ (j=1,2,\ldots ,m-1)$, we have

$$ \prod_{j=1}^{m}\lambda _{j}\geq \sum_{j=1}^{m} \frac{\lambda _{j}^{p_{j}}}{p _{j}}. $$

(2)

Lemma 2.2

([10])

Let $\mathbb{T}$be a time scale, $a, b\in \mathbb{T}$with $a< b$and $\sum_{j=1}^{m}\frac{1}{p_{j}}=1$. If $f_{j}(\delta )>0$, and $f_{j}\ (j=1,2,\ldots ,m)$is continuous real-valued function on $[\xi , \sigma ]_{\mathbb{T}}$, then

(1) for $p_{j}>1$, we have

$$ \int _{\xi }^{\sigma }\prod_{j=1}^{m} f_{j}(\delta )\Diamond _{\alpha } \delta \leq \prod _{j=1}^{m} \biggl( \int _{\xi }^{\sigma } f_{j}^{p_{j}}( \delta )\Diamond _{\alpha }\delta \biggr)^{1/p_{j}}, $$

(3)

(2) for $0< p_{m}<1, p_{j}<0\ (j=1,2,\ldots ,m-1)$, we have

$$ \int _{\xi }^{\sigma }\prod_{j=1}^{m} f_{j}(\delta )\Diamond _{\alpha } \delta \geq \prod _{j=1}^{m} \biggl( \int _{\xi }^{\sigma } f_{j}^{p_{j}}( \delta )\Diamond _{\alpha }\delta \biggr)^{1/p_{j}}. $$

(4)

Lemma 2.3

([17])

Let $f, g, h:\mathbb{T}\to \mathbb{R}$be ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}, p>1$with $q=p/(p-1)$. Then we have

$$ \int _{\xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert f(\delta )g(\delta ) \bigr\vert \Diamond \delta \leq \biggl( \int _{\xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert f(\delta ) \bigr\vert ^{p} \Diamond \delta \biggr)^{1/p} \biggl( \int _{\xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert g( \delta ) \bigr\vert ^{q}\Diamond \delta \biggr)^{1/q}. $$

(5)

Lemma 2.4

([17])

Let $f, g, h:\mathbb{T}\to \mathbb{R}$be ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}, p>1$with $q=p/(p-1)$. Then we have

$$\begin{aligned} & \biggl( \int _{\xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert f(\delta )+g(\delta ) \bigr\vert ^{p} \Diamond \delta \biggr)^{1/p} \\ &\quad \leq \biggl( \int _{\xi }^{\sigma } \bigl\vert h( \delta ) \bigr\vert \bigl\vert f(\delta ) \bigr\vert ^{p}\Diamond \delta \biggr)^{1/p}+ \biggl( \int _{ \xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert g(\delta ) \bigr\vert ^{p}\Diamond \delta \biggr)^{1/p}. \end{aligned}$$

(6)

Lemma 2.5

([17])

Let $f, g, h:\mathbb{T}\to \mathbb{R}$be ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}, 0< p<1$with $q=p/(p-1)$. If $g^{q}$is ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}$, then

$$ \int _{\xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert f(\delta )g(\delta ) \bigr\vert \Diamond \delta \geq \biggl( \int _{\xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert f(\delta ) \bigr\vert ^{p} \Diamond \delta \biggr)^{1/p} \biggl( \int _{\xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert g( \delta ) \bigr\vert ^{q}\Diamond \delta \biggr)^{1/q}. $$

(7)

Lemma 2.6

([17])

Let $f, g, h:\mathbb{T}\to \mathbb{R}$be ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}, 0< p<1$with $q=p/(p-1)$. Then we have

$$\begin{aligned} & \biggl( \int _{\xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert f(\delta )+g(\delta ) \bigr\vert ^{p} \Diamond \delta \biggr)^{1/p} \\ &\quad \geq \biggl( \int _{\xi }^{\sigma } \bigl\vert h( \delta ) \bigr\vert \bigl\vert f(\delta ) \bigr\vert ^{p}\Diamond \delta \biggr)^{1/p}+ \biggl( \int _{ \xi }^{\sigma } \bigl\vert h(\delta ) \bigr\vert \bigl\vert g(\delta ) \bigr\vert ^{p}\Diamond \delta \biggr)^{1/p}. \end{aligned}$$

(8)

3 Main results about diamond-α integral Hölder’s inequality

Now, based on Tian’s [18, 19] research results, we will give the following generalizations and refinements of the three-tuple diamond-α integral and n-tuple diamond-α integral Hölder inequality on time scales.

Theorem 3.1

Let $\mathbb{T}$be a time scale $a, b \in \mathbb{T}$with $a< b$and $\alpha _{kj}\in \mathbb{R}$ $(j=1, 2, \ldots , m, k=1,2, \ldots , s), \sum_{k=1}^{s}\frac{1}{p_{k}}=1, \sum_{k=1}^{s}\alpha _{kj}=0$. If $f_{j}(\delta )>0$, and $f_{j}\ (j=1,2,\ldots , m)$is a continuous real-valued function on $[\xi , \sigma ]_{\mathbb{T}}$, then

(1) for $p_{k}>1$, we have

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{j=1}^{m} f_{j}(\delta _{1},\delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad\leq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}}\prod _{j=1}^{m} f_{j}^{1+p_{k}\alpha _{kj}}( \delta _{1},\delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \Biggr)^{\frac{1}{p_{k}}}, \end{aligned} $$

(9)

(2) for $0< p_{s}<1, p_{k}<0\ (k=1,2,\ldots ,s-1)$, we have

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{j=1}^{m} f_{j}(\delta _{1},\delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad\geq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}}\prod _{j=1}^{m} f_{j}^{1+p_{k}\alpha _{kj}}( \delta _{1},\delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \Biggr)^{\frac{1}{p_{k}}}. \end{aligned} $$

(10)

Proof

(1) Set

$$ \begin{aligned} g_{k}(\delta _{1}, \delta _{2}, \delta _{3})= \Biggl(\prod _{j=1}^{m} f_{j} ^{1+p_{k} \alpha _{k j}}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr)^{1 / p_{k}}. \end{aligned} $$

Applying the assumptions $\sum_{k=1}^{s}\frac{1}{p_{k}}=1$ and $\sum_{k=1}^{s}\alpha _{kj}=0$, by computing, we can observe that

$$\begin{aligned} & \prod_{k=1}^{s} g_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \\ &\quad=g_{1} g _{2} \cdots g_{s} \\ &\quad= \Biggl(\prod_{j=1}^{m} f_{j}^{1+a_{1} \alpha _{1 j}}( \delta _{1}, \delta _{2}, \delta _{3}) \Biggr)^{1 / a_{1}} \Biggl( \prod_{j=1} ^{m} f_{j}^{1+a_{2} \alpha _{2 j}}( \delta _{1}, \delta _{2}, \delta _{3}) \Biggr)^{1 / a_{2}}\cdots \\ &\qquad{}\times \Biggl(\prod_{j=1}^{m} f_{j}^{1+a_{s} \alpha _{s_{j}}}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr)^{1 / a_{s}} \\ &\quad=\prod_{j=1}^{m} f_{j}^{1 / a_{1}+\alpha _{1 j}}(\delta _{1}, \delta _{2}, \delta _{3}) \prod _{j=1}^{m} f_{j}^{1 / a_{2}+\alpha _{2 j}}( \delta _{1}, \delta _{2}, \delta _{3}) \cdots \\ &\qquad{}\times \prod_{j=1}^{m} f_{j} ^{1 / a_{s}+\alpha _{s j}}(\delta _{1}, \delta _{2}, \delta _{3}) \\ &\quad= \prod_{j=1}^{m} f_{j}^{1 / a_{1}+1 / a_{2}+\cdots +1 / a_{s}+\alpha _{1 j}+\alpha _{2 j}+\cdots +\alpha _{s j}}(\delta _{1}, \delta _{2}, \delta _{3})=\prod _{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}). \end{aligned}$$

Hence, we obtain

$$ \begin{aligned}[b] &\int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad = \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \prod _{k=1} ^{s} g_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\Diamond _{\alpha } \delta _{3}. \end{aligned} $$

(11)

By the Hölder inequality (3), we find

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{k=1}^{s} g_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad\leq \prod_{k=1}^{s} \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} g_{k}^{p_{k}}( \delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{ \alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\Diamond _{\alpha } \delta _{3} \biggr)^{1 / p_{k}}. \end{aligned} $$

(12)

Substituting $g_{k}(\delta _{1},\delta _{2},\delta _{3})$ into the inequality (12) can be obtained

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{j=1}^{m} f_{j}(\delta _{1},\delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad\leq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \prod _{j=1}^{m} f_{j}^{1+p_{k} \alpha _{k j}}( \delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1} \Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \Biggr)^{\frac{1}{p_{k}}}. \end{aligned} $$

(2) After the same proof as inequality (9), we get

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{k=1}^{s} g_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad\geq \prod_{k=1}^{s} \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} g_{k}^{p_{k}}( \delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{ \alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\Diamond _{\alpha } \delta _{3} \biggr)^{1 / p_{k}}. \end{aligned} $$

(13)

Substituting $g_{k}(\delta _{1},\delta _{2}.\delta _{3})$ into the (12) can be obtained

$$\begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{j=1}^{m} f_{j}(\delta _{1},\delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad\geq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \prod _{j=1}^{m} f_{j}^{1+p_{k} \alpha _{k j}}( \delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1} \Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \Biggr)^{\frac{1}{p_{k}}}. \end{aligned}$$

The proof of Theorem 3.1 is accomplished. □

Corollary 3.2

Under the conditions of Theorem 3.1, let $s=m, \alpha _{kj}=-t/p_{k}$for $k \neq j$and $\alpha _{jj}=t(1-1/p_{j})$with $t \in \mathbb{R}$, then

(1) for $p_{k}>1$, we have the following inequality:

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad\leq \prod_{k=1}^{m} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \Biggl(\prod _{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr)^{1-t} \\ &\qquad{}\times \bigl(f_{k}^{p_{k}}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr)^{t} \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\Diamond _{\alpha } \delta _{3} \Biggr)^{1 / p_{k}}, \end{aligned} $$

(14)

(2) $0< p_{m}<1, p_{k}<0\ (k=1,2,\ldots , m-1)$, we have the following reverse inequality:

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad\geq \prod_{k=1}^{m} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \Biggl(\prod _{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr)^{1-t} \\ &\qquad{}\times \bigl(f_{k}^{p_{k}}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr)^{t} \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\Diamond _{\alpha } \delta _{3} \Biggr)^{1 / p_{k}}. \end{aligned} $$

(15)

On the basis of Theorem 3.1, we give the n-tuple diamond-α integral Hölder’s inequality on time scales.

Theorem 3.3

Let $\mathbb{T}$be a time scale $\xi ,\sigma \in \mathbb{T}$with $\xi <\sigma$and $\alpha _{kj}\in \mathbb{R}$ $(j=1, 2, \ldots , m, k=1,2, \ldots , s), \sum_{k=1}^{s}\frac{1}{p_{k}}=1, \sum_{k=1}^{s} \alpha _{kj}=0$. If $f_{j}(\delta )>0$, and $f_{j}\ (j=1,2,\ldots , m)$is a continuous real-valued function on $[\xi , \sigma ]_{\mathbb{T}}$, then

(1) for $p_{k}>1$, we have the following inequality:

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \prod _{j=1}^{m} f_{j}(\delta _{1},\delta _{2},\ldots , \delta _{n}) \Diamond _{\alpha } \delta _{1}\Diamond _{ \alpha } \delta _{2}\cdots \Diamond _{\alpha } \delta _{n} \\ &\quad\leq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}}\prod _{j=1}^{m} f_{j}^{1+p_{k} \alpha _{kj}}( \delta _{1},\delta _{2},\ldots , \delta _{n})\Diamond _{ \alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\cdots \Diamond _{ \alpha } \delta _{n} \Biggr)^{\frac{1}{p_{k}}}, \end{aligned} $$

(16)

(2) for $0< p_{s}<1, p_{k}<0\ (k=1,2,\ldots ,s-1)$, we have the following reverse inequality:

$$\begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \prod _{j=1}^{m} f_{j}(\delta _{1},\delta _{2},\ldots , \delta _{n}) \Diamond _{\alpha } \delta _{1}\Diamond _{ \alpha } \delta _{2}\cdots \Diamond _{\alpha } \delta _{n} \\ &\quad\geq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}}\prod _{j=1}^{m} f_{j}^{1+p_{k} \alpha _{kj}}( \delta _{1},\delta _{2},\ldots , \delta _{n})\Diamond _{ \alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\cdots \Diamond _{ \alpha } \delta _{n} \Biggr)^{\frac{1}{p_{k}}}. \end{aligned}$$

(17)

Proof

Similar to the proof of Theorem 3.1, we get the result of Theorem 3.3. □

Remark 3.4

The three-tuple diamond-$alpha$ inequalities in Theorem 3.1 and the n-tuple diamond-α inequalities in Theorem 3.3 are generalizations to Theorem 3.3 in Ref. [10].

Theorem 3.5

Let $\mathbb{T}$be a time scale, $\xi , \sigma \in \mathbb{T}$with $\xi <\sigma$and $r\in \mathbb{R}, \alpha _{kj}\in \mathbb{R}\ (j=1,2, \ldots ,m, k=1,2,\ldots ,s)$, $\sum_{k=1}^{s}\frac{1}{p_{k}}=r, \sum_{k=1}^{s}\alpha _{kj}=0$. If $f_{j}(\delta )>0$, and $f_{j}\ (j=1,2, \ldots ,m)$is a continuous real-valued function on $[\xi , \sigma ]_{ \mathbb{T}}$, then

(1) for $rp_{k}>1$, we have the following inequality:

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{j=1}^{m} f_{j}(\delta _{1},\delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad\leq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}}\prod _{j=1}^{m} f_{j}^{1+rp_{k}\alpha _{kj}}( \delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\Diamond _{\alpha } \delta _{3} \Biggr)^{\frac{1}{rp_{k}}}, \end{aligned} $$

(18)

(2) for $0< rp_{k}<1, rp_{k}<0\ (k=1, 2,\ldots , s-1)$, we have the following reverse inequality:

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \prod _{j=1}^{m} f_{j}(\delta _{1},\delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2} \Diamond _{\alpha } \delta _{3} \\ &\quad\geq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}}\prod _{j=1}^{m} f_{j}^{1+rp_{k}\alpha _{kj}}( \delta _{1}, \delta _{2}, \delta _{3}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\Diamond _{\alpha } \delta _{3} \Biggr)^{\frac{1}{rp_{k}}}. \end{aligned} $$

(19)

Proof

(1) According to $rp_{k}>1$ and $\sum_{k=1}^{s}\frac{1}{p_{k}}=r$, we get $\sum_{k=1}^{s}\frac{1}{rp_{k}}=1$. Then, by inequality (9), we immediately obtain the inequality (18).

(2) According to $0< rp_{k}<1, rp_{k}<0\ (k=1,2,\ldots ,s-1)$ and $\sum_{k=1}^{s}\frac{1}{p_{k}}=r$, we have $\sum_{k=1}^{s}\frac{1}{rp _{k}}=1$, by inequality (10), we immediately have the inequality (19). This completes the proof. □

Similarly, on the basis of Theorem 3.5, we give the n-tuple diamond-α integral Hölder’s inequality on time scales.

Theorem 3.6

Let $\mathbb{T}$be a time scale, $\xi , \sigma \in \mathbb{T}$with $\xi <\sigma$and $r\in \mathbb{R}, \alpha _{kj}\in \mathbb{R}\ (j=1,2, \ldots ,m, k=1,2,\ldots ,s)$, $\sum_{k=1}^{s}\frac{1}{p_{k}}=r, \sum_{k=1}^{s}\alpha _{kj}=0$. If $f_{j}(\delta )>0$, and $f_{j}\ (j=1,2, \ldots ,m)$is a continuous real-valued function on $[\xi , \sigma ]_{ \mathbb{T}}$, then

(1) for $rp_{k}>1$, we have the following inequality:

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \prod _{j=1}^{m} f_{j}(\delta _{1},\delta _{2},\ldots , \delta _{n}) \Diamond _{\alpha } \delta _{1}\Diamond _{ \alpha } \delta _{2}\cdots \Diamond _{\alpha } \delta _{n} \\ &\quad\leq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}}\prod _{j=1}^{m} f_{j}^{1+rp _{k}\alpha _{kj}}( \delta _{1},\delta _{2},\ldots , \delta _{n}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\cdots \Diamond _{\alpha } \delta _{n} \Biggr)^{\frac{1}{rp_{k}}}, \end{aligned} $$

(20)

(2) for $0< rp_{k}<1, rp_{k}<0\ (k=1, 2,\ldots , s-1)$, we have the following reverse inequality:

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \prod _{j=1}^{m} f_{j}(\delta _{1},\delta _{2},\ldots , \delta _{n}) \Diamond _{\alpha } \delta _{1}\Diamond _{ \alpha } \delta _{2}\cdots \Diamond _{\alpha } \delta _{n} \\ &\quad\geq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}}\prod _{j=1}^{m} f_{j}^{1+rp _{k}\alpha _{kj}}( \delta _{1},\delta _{2},\ldots , \delta _{n}) \Diamond _{\alpha } \delta _{1}\Diamond _{\alpha } \delta _{2}\cdots \Diamond _{\alpha } \delta _{n} \Biggr)^{\frac{1}{rp_{k}}}. \end{aligned} $$

(21)

Proof

Similar to the proof of Theorem 3.5, we get the result of Theorem 3.6. □

Remark 3.7

For the inequality of Theorem 3.4 in the Reference [10], we put forward Theorem 3.5 and Theorem 3.6 as the generalization results.

Theorem 3.8

Assume that $\mathbb{T}$is a time scale, $\xi , \sigma \in \mathbb{T}$with $\xi <\sigma$and $p_{k}>0, \alpha _{kj}\in \mathbb{R}\ (j=1,2,\ldots ,m, k=1,2,\ldots , s), \sum_{k=1}^{s}\frac{1}{p _{k}}=1, \sum_{k=1}^{s}\alpha _{kj}=0, f_{j},h:\mathbb{T}\to \mathbb{R}$. Ifhand $f_{j}$are ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}$, then the following assertions hold true.

(1) For $p_{k}>1$, one has

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad \leq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{1+p_{k} \alpha _{k j}} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \Biggr)^{1 / p_{k}}. \end{aligned} $$

(22)

(2) For $0< p_{s}<1, p_{k}<0\ (k=1,2,\ldots ,s-1), f_{j}^{1+p_{k}\alpha _{kj}}$is ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}$, one has

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad \geq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{1+p_{k} \alpha _{k j}} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \Biggr)^{1 / p_{k}}. \end{aligned} $$

(23)

Proof

(1) Let

$$ \begin{aligned} g_{k}(\delta _{1}, \delta _{2}, \delta _{3})= \Biggl(\prod _{j=1}^{m} f_{j} ^{1+p_{k}\alpha _{kj}(\delta _{1}, \delta _{2}, \delta _{3})} \Biggr)^{1/p _{k}}. \end{aligned} $$

Based on the assumptions $\sum_{k=1}^{s}\frac{1}{p_{k}}=1$ and $\sum_{k=1}^{s}\alpha _{kj}=0$, from a direct computation, it is obvious to show that

$$\begin{aligned} &\prod_{k=1}^{s} g_{k}(\delta _{1}, \delta _{2}, \delta _{3})\\ &\quad=g_{1} g _{2} \cdots g_{s} \\ &\quad= \Biggl(\prod_{j=1}^{m} f_{j}^{1+a_{1} \alpha _{1 j}}( \delta _{1}, \delta _{2}, \delta _{3}) \Biggr)^{1 / a_{1}} \Biggl( \prod_{j=1} ^{m} f_{j}^{1+a_{2} \alpha _{2 j}}( \delta _{1}, \delta _{2}, \delta _{3}) \Biggr)^{1 / a_{2}}\cdots \\ &\qquad{}\times\Biggl(\prod_{j=1}^{m} f_{j}^{1+a_{s} \alpha _{s_{j}}}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr)^{1 / a_{s}} \\ &\quad=\prod_{j=1}^{m} f_{j}^{1 / a_{1}+\alpha _{1 j}}(\delta _{1}, \delta _{2}, \delta _{3}) \prod _{j=1}^{m} f_{j}^{1 / a_{2}+\alpha _{2 j}}( \delta _{1}, \delta _{2}, \delta _{3}) \cdots \\ &\qquad{}\times \prod_{j=1}^{m} f_{j} ^{1 / a_{s}+\alpha _{s j}}(\delta _{1}, \delta _{2}, \delta _{3}) \\ &\quad= \prod_{j=1}^{m} f_{j}^{1 / a_{1}+1 / a_{2}+\cdots +1 / a_{s}+\alpha _{1 j}+\alpha _{2 j}+\cdots +\alpha _{s j}}(\delta _{1}, \delta _{2}, \delta _{3})=\prod _{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}). \end{aligned}$$

From the above result, we can obtain

$$ \begin{aligned} \prod_{k=1}^{s} g_{k}(\delta _{1}, \delta _{2}, \delta _{3})=\prod_{j=1} ^{m}f_{j}(\delta _{1}, \delta _{2}, \delta _{3}). \end{aligned} $$

Hence, we have

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad= \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} g_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}. \end{aligned} $$

It follows from Hölder’s inequality (5) that

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} g_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \prod_{k=1}^{s} \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p _{k}} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p _{k}}. \end{aligned} $$

Thus, we have

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad \leq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{1+p_{k} \alpha _{k j}} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \Biggr)^{1 / p_{k}}. \end{aligned} $$

(2) The proof of inequality (23) is similar to the proof of inequality (22), we have

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} g_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \prod_{k=1}^{s} \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p _{k}} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p _{k}}. \end{aligned} $$

Thus, we have

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ & \quad\geq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{1+p_{k} \alpha _{k j}} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \Biggr)^{1 / p_{k}}. \end{aligned} $$

Thus, the proof of Theorem 3.8 is completed. □

Corollary 3.9

Under the assumptions of Theorem 3.8, taking $s=m, \alpha _{kj}=-t/p_{k}$for $j \neq k$and $\alpha _{kk}=t(1-1/p_{k})$with $t \in \mathbb{R}$, the following assertions hold true.

(1) For $p_{k}>1$, one has

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \prod_{k=1}^{m} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl(\prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggr)^{1-t} \\ &\qquad{}\times \bigl( \bigl\vert f_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p_{k}}\bigr)^{t}\Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3} \Biggr)^{1/p_{k}}. \end{aligned} $$

(2) For $0< p_{m}<1, p_{k}<0\ (k=1,2,\ldots ,m-1)$, one has

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \prod_{k=1}^{m} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl(\prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggr)^{1-t} \\ &\qquad{}\times \bigl( \bigl\vert f_{k}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p_{k}}\bigr)^{t}\Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3} \Biggr)^{1/p_{k}}. \end{aligned} $$

Theorem 3.10

Assume that $\mathbb{T}$is a time scale, $\xi , \sigma \in \mathbb{T}$with $\xi <\sigma$and $p_{k}>0, \alpha _{kj}\in \mathbb{R}\ (j=1,2,\ldots ,m, k=1,2,\ldots , s), \sum_{k=1}^{s}\frac{1}{p _{k}}=1, \sum_{k=1}^{s}\alpha _{kj}=0, f_{j},h:\mathbb{T}\to \mathbb{R}$. Ifhand $f_{j}$are ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}$, then the following assertions hold true.

(1) For $p_{k}>1$, one has

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \Biggr\vert \Diamond \delta _{1} \Diamond \delta _{2} \cdots \Diamond \delta _{n} \\ &\quad \leq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}} ^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \\ &\qquad{}\times\prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{1+p_{k} \alpha _{k j}} \Diamond \delta _{1}\Diamond \delta _{2} \cdots \Diamond \delta _{n} \Biggr)^{1 / p_{k}}. \end{aligned} $$

(24)

(2) For $0< p_{s}<1, p_{k}<0\ (k=1,2,\ldots ,s-1), f_{j}^{1+p_{k}\alpha _{kj}}$is ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}$, one has

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \Biggr\vert \Diamond \delta _{1} \Diamond \delta _{2} \cdots \Diamond \delta _{n} \\ & \quad\geq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}} ^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \\ &\qquad{}\times\prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{1+p_{k} \alpha _{k j}} \Diamond \delta _{1}\Diamond \delta _{2} \cdots \Diamond \delta _{n} \Biggr)^{1 / p_{k}}. \end{aligned} $$

(25)

Proof

Similar to the proof of Theorem 3.8, we get the result of Theorem 3.10. □

Remark 3.11

The inequalities in Theorem 3.8 and Theorem 3.10 are the result of generalization of Theorem 4.1 in Ref. [17].

Theorem 3.12

Assume that $\mathbb{T}$is a time scale, $\xi , \sigma \in \mathbb{T}$with $\xi <\sigma$and $p_{k}>0, r\in \mathbb{R}, \alpha _{kj}\in \mathbb{R}\ (j=1,2,\ldots ,m, k=1,2,\ldots ,s), \sum_{k=1} ^{s}\frac{1}{p_{k}}=r, \sum_{k=1}^{s}\alpha _{kj}=0, f_{j},h: \mathbb{T}\to \mathbb{R}$. If $f_{j}$and h are ◊-integrable on $[\xi , \sigma ]_{T}$, then the following assertions hold true.

(1) For $rp_{k}>1$, one has

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{1+rp _{k}\alpha _{kj}}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \Biggr)^{1/rp_{k}}. \end{aligned} $$

(26)

(2) For $0< rp_{k}<1, rp_{k}<0\ (k=1,2,\ldots , s-1), f_{j}^{1+rp_{k} \alpha _{kj}}$is ◊-integrable on $[\xi , \sigma ]_{ \mathbb{T}}$, one has

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \Biggr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{1+rp _{k}\alpha _{kj}}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \Biggr)^{1/rp_{k}}. \end{aligned} $$

(27)

Proof

(1) Since $rp_{k}>1$ and $\sum_{k=1}^{s}\frac{1}{rp_{k}}=1$. Then by inequality (22) we can obtain inequality (26).

(2) Since $0< rp_{s}<1, rp_{k}<0$ and $\sum_{k=1}^{s}\frac{1}{rp_{k}}=1$, by inequality (23), we can obtain inequality (27).

The proof of Theorem 3.12 is completed. □

Theorem 3.13

Assume that $\mathbb{T}$is a time scale, $\xi , \sigma \in \mathbb{T}$with $\xi <\sigma$and $p_{k}>0, r\in \mathbb{R}, \alpha _{kj}\in \mathbb{R}\ (j=1,2,\ldots ,m, k=1,2,\ldots ,s), \sum_{k=1} ^{s}\frac{1}{p_{k}}=r, \sum_{k=1}^{s}\alpha _{kj}=0, f_{j},h: \mathbb{T}\to \mathbb{R}$. If $f_{j}$and h are ◊-integrable on $[\xi , \sigma ]_{T}$, then the following assertions hold true.

(1) For $rp_{k}>1$, one has

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \Biggr\vert \Diamond \delta _{1} \Diamond \delta _{2} \cdots \Diamond \delta _{n} \\ &\quad\leq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}} ^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \\ &\qquad{}\times\prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{1+rp_{k}\alpha _{kj}}\Diamond \delta _{1}\Diamond \delta _{2} \cdots \Diamond \delta _{n} \Biggr)^{1/rp_{k}}. \end{aligned} $$

(28)

(2) For $0< rp_{k}<1, rp_{k}<0\ (k=1,2,\ldots , s-1), f_{j}^{1+rp_{k} \alpha _{kj}}$is ◊-integrable on $[\xi , \sigma ]_{ \mathbb{T}}$, one has

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \Biggl\vert \prod_{j=1}^{m} f_{j}(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \Biggr\vert \Diamond \delta _{1} \Diamond \delta _{2} \cdots \Diamond \delta _{n} \\ &\quad\geq \prod_{k=1}^{s} \Biggl( \int _{\xi _{1}} ^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \\ &\qquad{}\times\prod_{j=1}^{m} \bigl\vert f_{j}(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{1+rp_{k}\alpha _{kj}}\Diamond \delta _{1}\Diamond \delta _{2} \cdots \Diamond \delta _{n} \Biggr)^{1/rp_{k}}. \end{aligned} $$

(29)

Proof

Similar to the proof of Theorem 3.12, we get the result of Theorem 3.13. □

Theorem 3.14

Let $f, g, h:\mathbb{T}\to \mathbb{R}$be ◊-integrable on $[\xi , \sigma ]_{T}$, and $s, t \in \mathbb{R}$, and let $p=(s-t)/(1-t), q=(s-t)/(s-1)$.

(1) If $s<1<t$or $s>1>t$, then

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{sp}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p^{2}} \\ &\qquad{} \times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{tq}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/q^{2}} \\ &\qquad{} \times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{tp}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\qquad{}\times \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{sq}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/pq}. \end{aligned} $$

(30)

(2) If $s>t>1$or $s< t<1$; $t>s>1$or $t< s<1$, then

$$\begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{sp}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p^{2}} \\ & \qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{tq}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/q^{2}} \\ &\qquad{} \times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{tp}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\qquad{}\times \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{sq}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/pq}. \end{aligned}$$

(31)

Proof

(1) Let $p=\frac{s-t}{1-t}$ and in view of $s<1<t$ or $s>1>t$, we have

$$ \begin{aligned} p=\frac{s-t}{1-t}>1, \end{aligned} $$

by Hölder’s inequality (5) with indices $\frac{s-t}{1-t}$ and $\frac{s-t}{s-1}$, we have

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \vert h \vert \vert fg \vert \Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \\ &\quad= \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert fg \vert ^{s(1-t)/(s-t)} \vert fg \vert ^{t(s-1)/(s-t)} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert fg \vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(1-t)/(s-t)} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \vert h \vert \vert fg \vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(s-1)/(s-t)}. \end{aligned} $$

On the other hand, from Hölder’s inequality (5) again for $p=\frac{s-t}{1-t}>1$, it follows that the following two inequalities are true:

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \vert h \vert \vert fg \vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}} ^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert f \vert ^{s(s-t)/(1-t)}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(1-t)/(s-t)} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert g \vert ^{s(s-t)/(s-1)}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(s-1)/(s-t)} \end{aligned} $$

and

$$\begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \vert h \vert \vert fg \vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}} ^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert f \vert ^{t(s-t)/(1-t)}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(1-t)/(s-t)} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert g \vert ^{t(s-t)/(s-1)}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(s-1)/(s-t)}. \end{aligned}$$

Thus, we have

$$\begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{sp}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p^{2}} \\ & \qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{tq}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/q^{2}} \\ &\qquad{} \times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{tp}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\qquad{}\times \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{sq}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/pq}. \end{aligned}$$

(2) Let $p=\frac{s-t}{1-t}$ and in view of $s>t>1$ or $s< t<1$, we have

$$ \begin{aligned} p=\frac{s-t}{1-t}< 0 \end{aligned} $$

and $t>s>1$ or $t< s<1$, we have $0<\frac{s-t}{1-t}<1$, by the reverse Hölder inequality (6) with indices $\frac{s-t}{1-t}$ and $\frac{s-t}{s-1}$, we have

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \vert h \vert \vert fg \vert \Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \\ &\quad= \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert fg \vert ^{s(1-t)/(s-t)} \vert fg \vert ^{t(s-1)/(s-t)} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert fg \vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(1-t)/(s-t)} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \vert h \vert \vert fg \vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(s-1)/(s-t)}. \end{aligned} $$

On the other hand, from Hölder’s inequality (6) again for $0< p=\frac{s-t}{1-t}<1$ or $p=\frac{s-t}{1-t}<0$, it follows that the following two inequalities are true:

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \vert h \vert \vert fg \vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}} ^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert f \vert ^{s(s-t)/(1-t)}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(1-t)/(s-t)} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert g \vert ^{s(s-t)/(s-1)}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(s-1)/(s-t)} \end{aligned} $$

and

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \vert h \vert \vert fg \vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}} ^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert f \vert ^{t(s-t)/(1-t)}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(1-t)/(s-t)} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h \vert \vert g \vert ^{t(s-t)/(s-1)}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(s-1)/(s-t)}. \end{aligned} $$

Thus, we have

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{sp}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p^{2}} \\ &\qquad{} \times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{tq}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/q^{2}} \\ & \qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{tp}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\qquad{}\times \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{sq}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/pq}. \end{aligned} $$

Thus, the proof of Theorem 3.14 is completed. □

Theorem 3.15

Let $f, g, h:\mathbb{T}\to \mathbb{R}$be ◊-integrable on $[\xi , \sigma ]_{T}$, and $s, t \in \mathbb{R}$, and let $p=(s-t)/(1-t), q=(s-t)/(s-1)$.

(1) If $s<1<t$or $s>1>t$, then

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1},\delta _{2}, \ldots , \delta _{n})g( \delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \Diamond \delta _{1} \Diamond \delta _{2}\cdots \Diamond \delta _{n} \\ &\quad\leq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{sp}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{1/p^{2}} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h( \delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{tq}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{1/q^{2}} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{tp}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \\ &\qquad{}\times \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert \bigl\vert g(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{sq}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{1/pq}. \end{aligned} $$

(32)

(2) If $s>t>1$or $s< t<1$; $t>s>1$or $t< s<1$, then

$$\begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1},\delta _{2}, \ldots , \delta _{n})g( \delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \Diamond \delta _{1} \Diamond \delta _{2}\cdots \Diamond \delta _{n} \\ &\quad\geq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{sp}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{1/p^{2}} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h( \delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{tq}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{1/q^{2}} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{tp}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \\ &\qquad{}\times \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert \bigl\vert g(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{sq}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{1/pq}. \end{aligned}$$

(33)

Proof

Similar to the proof of Theorem 3.14, we get the result of Theorem 3.15. □

Remark 3.16

For the inequality of Theorem 5.1 in the Reference [17], we put forward Theorem 3.14 and Theorem 3.15 as the generalization results.

4 Main results about diamond-alpha integral Minkowski’s inequality

Next, we give some generalizations of diamond-α integral Minkowski’s inequality in the following theorems.

Theorem 4.1

Let $f, g, h: \mathbb{T}\to \mathbb{R}$be $\Diamond -integrable$on $[\xi , \sigma ]_{\mathbb{T}}, p>0, s,t\in \mathbb{R}$, and $s\neq t$.

(1) Let $p, s, t\in \mathbb{R}$be different such that $s,t>1$and $(s-t)/(p-t)>1$. Then

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{s}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{\frac{1}{s}} \biggr]^{s(p-t)/(s-t)} \\ &\qquad{} \times \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{t}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{\frac{1}{t}} \biggr]^{t(s-p)/(s-t)}. \end{aligned} $$

(34)

(2) Let $p, s, t\in \mathbb{R}$be different such that $0< s<1, 0<t<1$and $(s-t)/(p-t)<1$. Then

$$ \begin{aligned}[b] & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{s}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{\frac{1}{s}} \biggr]^{s(p-t)/(s-t)} \\ &\qquad{} \times \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{t}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{\frac{1}{t}} \biggr]^{t(s-p)/(s-t)}. \end{aligned} $$

(35)

Proof

(1) We have $(s-t)/(p-t)>1$, and

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad= \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl( \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s} \bigr)^{(p-t)/(s-t)} \\ &\qquad{}\times \bigl( \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t} \bigr)^{(s-p)/(s-t)} \Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3}, \end{aligned} $$

by using Hölder’s inequality (5) with indices $(s-t)/(p-t)$ and $(s-t)/(s-p)$, we have

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(p-t)/(s-t)} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(s-p)/(s-t)}. \end{aligned} $$

On the other hand, by using Minkowski’s inequality (6) for $s>1$ and $t>1$, respectively, we can see that the following assertions hold true:

$$ \begin{aligned} & \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{s}} \\ &\quad\leq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{ \frac{1}{s}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{ \sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{s}} \end{aligned} $$

and

$$ \begin{aligned} & \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{t}} \\ &\quad\leq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{ \frac{1}{t}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{ \sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{t}}. \end{aligned} $$

So we get the result

$$\begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{s}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{\frac{1}{s}} \biggr]^{s(p-t)/(s-t)} \\ &\qquad{} \times \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{t}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{\frac{1}{t}} \biggr]^{t(s-p)/(s-t)}. \end{aligned}$$

(2) We have $(s-t)/(p-t)<1$ and

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad= \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl( \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s} \bigr)^{(p-t)/(s-t)} \\ &\qquad{}\times \bigl( \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t} \bigr)^{(s-p)/(s-t)} \Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3}, \end{aligned} $$

by using Hölder’s inequality (7) with indices $(s-t)/(p-t)$ and $(s-t)/(s-p)$, we have

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(p-t)/(s-t)} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{(s-p)/(s-t)}. \end{aligned} $$

On the other hand, by using Minkowski’s inequality (8) for $s>1$ and $t>1$, respectively, we can see that the following assertions hold true

$$ \begin{aligned} & \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{s}} \\ &\quad\geq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{ \frac{1}{s}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{ \sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{s}} \end{aligned} $$

and

$$ \begin{aligned} & \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{t}} \\ &\quad\geq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{ \frac{1}{t}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{ \sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{t}}. \end{aligned} $$

So we get the result

$$\begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{s}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{\frac{1}{s}} \biggr]^{s(p-t)/(s-t)} \\ & \qquad{}\times \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{t}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{\frac{1}{t}} \biggr]^{t(s-p)/(s-t)}. \end{aligned}$$

 □

Remark 4.2

(1) Under the conditions of Theorem 4.1, for $p>1$, letting $s=p+\varepsilon , t=p-\varepsilon$, when $p, s, t$ are different, $s,t>1$, and letting $\varepsilon \to 0$, we obtain

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\leq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{p}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{\frac{1}{p}}. \end{aligned} $$

(2) Under the conditions of Theorem 4.1, for $0< p<1$, letting $s=p+\varepsilon , t=p-\varepsilon$, when $p, s, t$ are different, $0< s,t<1$, and letting $\varepsilon \to 0$, we obtain

$$ \begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &\quad\geq \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{\frac{1}{p}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{\frac{1}{p}}. \end{aligned} $$

Now, on the basis of Theorem 4.1, we give the n-tuple diamond-α integral Minkowski’s inequality on time scales.

Theorem 4.3

Let $f, g, h: \mathbb{T}\to \mathbb{R}$be ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}, p>0, s,t\in \mathbb{R}$, and $s\neq t$.

(1) Let $p, s, t\in \mathbb{R}$be different such that $s,t>1$and $(s-t)/(p-t)>1$. Then

$$\begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1},\delta _{2}, \ldots , \delta _{n})+g(\delta _{1}, \delta _{2},\ldots , \delta _{n}) \bigr\vert ^{p}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \\ &\quad\leq \biggl[ \biggl( \int _{\xi _{1}}^{ \sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{\frac{1}{s}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \\ &\qquad{}\times \bigl\vert g(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{ \frac{1}{s}} \biggr]^{s(p-t)/(s-t)} \\ &\qquad{}\times \biggl[ \biggl( \int _{\xi _{1}} ^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{\frac{1}{t}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \\ &\qquad{}\times\bigl\vert g(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{ \frac{1}{t}} \biggr]^{t(s-p)/(s-t)}. \end{aligned}$$

(36)

(2) Let $p, s, t\in \mathbb{R}$be different such that $0< s<1, 0<t<1$and $(s-t)/(p-t)<1$. Then

$$\begin{aligned} & \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1},\delta _{2}, \ldots , \delta _{n})+g(\delta _{1}, \delta _{2},\ldots , \delta _{n}) \bigr\vert ^{p}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \\ &\quad\geq \biggl[ \biggl( \int _{\xi _{1}}^{ \sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{\frac{1}{s}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \\ &\qquad{}\times\bigl\vert g(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{s}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{ \frac{1}{s}} \biggr]^{s(p-t)/(s-t)} \\ &\qquad{}\times \biggl[ \biggl( \int _{\xi _{1}} ^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{\frac{1}{t}} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \bigl\vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \bigr\vert \\ &\qquad{}\times\bigl\vert g(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \bigr\vert ^{t}\Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n} \biggr)^{ \frac{1}{t}} \biggr]^{t(s-p)/(s-t)}. \end{aligned}$$

(37)

Proof

Similar to the proof of Theorem 4.1, we get the result of Theorem 4.3. □

Remark 4.4

Aiming at the diamond-α integral Minkowski’s inequality proposed by Theorem 3.5 in Ref. [17], we generalize it in this paper and obtain the three-tuple and n-tuple diamond-α inequalities (34)–(37).

Theorem 4.5

Let $f,g,h: \mathbb{T}\to \mathbb{R}$and $0< r<1<p$. If $f,g$andhare ◊-integrable on $[\xi ,\sigma ]_{\mathbb{T}}$, then

$$ \begin{aligned}[b] & \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{ \int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}} ^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)} \\ &\quad\leq \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)} \\ &\qquad{}+ \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)}. \end{aligned} $$

(38)

Proof

From inequality (5) and inequality (6), we have

$$\begin{aligned} & \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/(p-r)} \\ &\quad\leq \biggl( \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p} \biggr)^{p/(p-r)} \\ &\quad= \biggl( \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1/p} \\ &\qquad{} \times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p} \\ &\qquad{}+ \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1/p} \\ &\qquad{}\times \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p} \biggr)^{p/(p-r)} \\ &\quad\leq \biggl( \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1/(p-r)} \\ &\qquad{}+ \biggl(\frac{ \int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}} ^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1/(p-r)} \biggr) \\ &\qquad{}\times \biggl( \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/r} \\ &\qquad{}+ \biggl( \int _{\xi _{1}} ^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/r} \biggr)^{r/(p-r)}. \end{aligned}$$

From inequality (6), we get

$$\begin{aligned} & \biggl( \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/r} \\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{ \sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/r} \biggr)^{r} \\ &\quad\leq \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3}. \end{aligned}$$

Hence, we have

$$\begin{aligned} & \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{ \int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}} ^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)} \\ &\quad\leq \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)} \\ &\qquad{}+ \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)}. \end{aligned}$$

The proof of Theorem 4.5 is completed. □

Next, on the basis of Theorem 4.5, we give the n-tuple diamond-α integral Minkowski’s inequality on time scales.

Theorem 4.6

Let $f,g,h: \mathbb{T}\to \mathbb{R}$and $0< r<1<p$. If $f,g$andhare ◊-integrable on $[\xi ,\sigma ]_{\mathbb{T}}$, then

$$\begin{aligned} & \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1}, \delta _{2},\ldots , \delta _{n}) \vert \vert f(\delta _{1}, \delta _{2},\ldots , \delta _{n})+g(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n}}{\int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \vert \vert f(\delta _{1}, \delta _{2}, \ldots , \delta _{n})+g(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n}} \biggr)^{1 /(p-r)} \\ &\quad \leq \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1}, \delta _{2},\ldots ,\delta _{n}) \vert \vert f(\delta _{1}, \delta _{2},\ldots , \delta _{n}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1}, \delta _{2}, \ldots ,\delta _{n}) \vert \vert f(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n}} \biggr)^{1 /(p-r)} \\ &\qquad{} + \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1}, \delta _{2},\ldots , \delta _{n}) \vert \vert g(\delta _{1}, \delta _{2}, \ldots , \delta _{n}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2} \cdots \Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}} ^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1}, \delta _{2},\ldots , \delta _{n}) \vert \vert g(\delta _{1}, \delta _{2}, \ldots ,\delta _{n}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n}} \biggr)^{1 /(p-r)}. \end{aligned}$$

(39)

Proof

Similar to the proof of Theorem 4.5, we get the result of Theorem 4.6. □

Remark 4.7

The inequalities in Theorem 4.5 and Theorem 4.6 are generalized results for Theorem 3.6 in Ref. [17].

Theorem 4.8

Let $f,g,h: \mathbb{T}\to \mathbb{R}$and $p\leq 0\leq r$. If $f,g, f^{p},g^{p}$andhare ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}$, then

$$ \begin{aligned}[b] & \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{ \int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}} ^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)} \\ &\quad\geq \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)} \\ &\qquad{}+ \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)}. \end{aligned} $$

(40)

Proof

Let $\alpha _{1}\geq 0, \alpha _{2}\geq 0, \beta _{1}>0$, and $\beta _{2}>0$, and $-1<\lambda <0$, using the following Radon inequality:

$$ \sum_{k=1}^{n} \frac{a_{k}^{p}}{b_{k}^{p-1}} \leq \frac{ (\sum_{k=1}^{n} a_{k} )^{p}}{ (\sum_{k=1}^{n} b_{k} )^{p-1}},\quad a_{k} \geq 0, b_{k}>0,0< p< 1, $$

we have

$$ \frac{\alpha _{1}^{\lambda +1}}{\beta _{1}^{\lambda }}+\frac{\alpha _{2} ^{\lambda +1}}{\beta _{2}^{\lambda }} \leq \frac{ (\alpha _{1}+ \alpha _{2} )^{\lambda +1}}{ (\beta _{1}+\beta _{2} )^{ \lambda }}. $$

Let

$$\begin{aligned} &\alpha _{1}= \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert f \vert ^{p} \Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1 / p}, \\ &\beta _{1}= \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert f \vert ^{r} \Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1 / r},\\ & \alpha _{2}= \biggl( \int _{\xi _{1}} ^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert g \vert ^{p} \Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1 / p}, \\ &\beta _{2}= \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert g \vert ^{r} \Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1 / r}, \end{aligned}$$

and let $\lambda =\frac{r}{p-r}$, it follows that

$$ \begin{aligned} &\frac{\alpha _{1}^{\lambda +1}}{\beta _{1}^{\lambda }}+\frac{\alpha _{2} ^{\lambda +1}}{\beta _{2}^{\lambda }} \\ &\quad= \frac{ (\int _{\xi _{1}}^{ \sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h( \delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} )^{(\lambda +1) / p}}{ ( \int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}} ^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} )^{\lambda / r}} \\ &\qquad{}+\frac{ (\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} )^{(\lambda +1) / p}}{ (\int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} )^{\lambda / r}} \\ &\quad= \biggl(\frac{ \int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}} ^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{ \sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h( \delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1/(p-r)} \\ &\qquad{}+ \biggl(\frac{ \int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}} ^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{ \sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h( \delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1/(p-r)} \\ &\quad\leq \frac{ ( \alpha _{1}+\alpha _{2} )^{\lambda +1}}{ (\beta _{1}+\beta _{2} )^{\lambda }} \\ &\quad= \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert f \vert ^{p}\Diamond \delta _{1} \Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{1/p} \\ &\qquad{} +\biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert g \vert ^{p}\Diamond \delta _{1} \Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{1/p} \biggr]^{p/(p-r)} \\ & \qquad\Big/ \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert f \vert ^{r}\Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/r} \\ &\qquad{} +\biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert g \vert ^{r} \Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/r} \biggr]^{r/(p-r)}. \end{aligned} $$

Since $-1<\lambda =\frac{r}{p-r}<0$, we may assume $p<0<r$, and $0< r\leq 1$, we obtain

$$\begin{aligned} & \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert f \vert ^{r} \Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1 / r}\\ &\qquad{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert g \vert ^{r} \Diamond \delta _{1} \Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{1 / r} \biggr]^{r} \\ &\quad\leq \int _{\xi _{1}} ^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \vert f+g \vert ^{r} \Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3}. \end{aligned}$$

For $p<0$, we obtain

$$ \begin{aligned} \sum_{k=1}^{n} m_{k}^{p} n_{k}^{1 / p-1} \geq \Biggl(\sum_{k=1}^{n} m _{k} \Biggr)^{p} \Biggl(\sum _{k=1}^{n} n_{k}^{1 / p} \Biggr)^{1-p}. \end{aligned} $$

Assume that $f(\delta )$ and $g(\delta )$ are nonzero, let

$$\begin{aligned} M={}& \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}, \\ N={}& \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}, \\ W={}& \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \biggr)^{1/p} \\ &{}+ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p} \\ ={}&M^{1/p}+N^{1/p}. \end{aligned}$$

From the above inequality, we have

$$ \begin{aligned} W={}&M^{1/p}+N^{1/p} \\ ={}&M^{1/p-1} \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ &{}+N^{1/p-1} \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3} \\ ={}& \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{ \sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl( \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}M^{1/p-1} \\ &{}+ \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}N^{1/p-1} \bigr) \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ \geq{}& \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}} ^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \\ &{} \times \bigl(M^{1/p}+N^{1/p} \bigr)^{1-p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \\ ={}& \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}W^{1-p} \Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3} \\ ={}&W^{1-p} \int _{\xi _{1}}^{ \sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}\Diamond \delta _{1} \Diamond \delta _{2}\Diamond \delta _{3}. \end{aligned} $$

That is,

$$ \begin{aligned} W\geq W^{1-p} \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}. \end{aligned} $$

Hence, we have

$$ \begin{aligned} W^{p}\geq \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}\Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}. \end{aligned} $$

Based on the above inequality, we obtain

$$ \begin{aligned} & \biggl[ \biggl( \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p} \\ &\qquad{}+ \biggl( \int _{\xi _{1}} ^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h( \delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3} \biggr)^{1/p} \biggr]^{p} \\ &\quad\geq \int _{\xi _{1}}^{\sigma _{1}} \int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \bigl\vert h(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert \bigl\vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \bigr\vert ^{p}\Diamond \delta _{1}\Diamond \delta _{2} \Diamond \delta _{3}. \end{aligned} $$

Through the above inequalities, we finally get the result

$$ \begin{aligned} & \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f( \delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{ \int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}} ^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f(\delta _{1}, \delta _{2}, \delta _{3})+g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)} \\ &\quad\leq \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert f(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)} \\ &\qquad{}+ \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g( \delta _{1}, \delta _{2}, \delta _{3}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\int _{\xi _{3}}^{\sigma _{3}} \vert h(\delta _{1}, \delta _{2}, \delta _{3}) \vert \vert g(\delta _{1}, \delta _{2}, \delta _{3}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\Diamond \delta _{3}} \biggr)^{1 /(p-r)}. \end{aligned} $$

Thus, Theorem 4.8 is completely proved. □

Theorem 4.9

Let $f,g,h: \mathbb{T}\to \mathbb{R}$and $0< r<1<p$. If f, g and h are ◊-integrable on $[\xi , \sigma ]_{\mathbb{T}}$, then we have the following assertion

$$ \begin{aligned}[b] & \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}} \cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \vert \vert f(\delta _{1},\delta _{2},\ldots , \delta _{n})+g(x_{1},x _{2}\cdots ,x_{n}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n}}{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \vert \vert f(\delta _{1},\delta _{2},\ldots , \delta _{n})+g( \delta _{1},\delta _{2},\ldots , \delta _{n}) \vert ^{r} \Diamond \delta _{1} \Diamond \delta _{2}\cdots \Diamond \delta _{n}} \biggr)^{1 /(p-r)} \\ & \quad\leq \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \vert \vert f(\delta _{1},\delta _{2},\ldots , \delta _{n}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n}}{ \int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \vert \vert f(\delta _{1},\delta _{2},\ldots , \delta _{n}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n}} \biggr)^{1 /(p-r)} \\ &\qquad{}+ \biggl(\frac{\int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1},\delta _{2}, \ldots , \delta _{n}) \vert \vert g(\delta _{1},\delta _{2},\ldots , \delta _{n}) \vert ^{p} \Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n}}{ \int _{\xi _{1}}^{\sigma _{1}}\int _{\xi _{2}}^{\sigma _{2}}\cdots \int _{\xi _{n}}^{\sigma _{n}} \vert h(\delta _{1},\delta _{2},\ldots , \delta _{n}) \vert \vert g(\delta _{1},\delta _{2},\ldots , \delta _{n}) \vert ^{r} \Diamond \delta _{1}\Diamond \delta _{2}\cdots \Diamond \delta _{n}} \biggr)^{1 /(p-r)}. \end{aligned} $$

(41)

Proof

Similar to the proof of Theorem 4.8, we get the result of Theorem 4.9. □

Remark 4.10

For the inequality of Theorem 3.7 in Ref. [17], we generalize it in this paper and obtain the generalized inequalities in Theorem 4.8 and Theorem 4.9.