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On Cauchy–Schwarz inequality for N-tuple diamond-alpha integral

  • Xi-Mei Hu
  • Jing-Feng TianEmail author
  • Yu-Ming Chu
  • Yan-Xia Lu
Open Access
Research

Abstract

In this paper, we present some new Cauchy–Schwarz inequalities for N-tuple diamond-alpha integral on time scales. The obtained results improve and generalize some Cauchy–Schwarz type inequalities given by many authors.

Keywords

Cauchy–Schwarz inequality Diamond-alpha integral Time scales 

MSC

26D15 

1 Introduction

To unify discrete and continuous analysis and generalize the discrete and continuous theories to cases “in between”, Stefan Hilger in [1] initiated the notions of a time scale and a delta derivative of a function defined on the time scale. The author then presented the calculus on time scales. If the time scale is an interval, the calculus is reduced to the classical calculus; if the time scale is discrete, the calculus is reduced to the calculus of finite differences. Since then, research in the area of the theory of time scales introduced by Stefan Hilger has exceeded by far a thousand publications, and numerous applications to all branches of science, such as operations research, engineering, economics, physics, finance, statistics, and biology, have been proposed. For more details on time scales theory, the interested readers may consult [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and the references therein.

As we all know, inequality plays a very important and basic role in all mathematic areas, and it is also an indispensable and basic tool in engineering technology (see [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]). The classic Cauchy–Schwarz inequality is an important cornerstone in some branches of mathematic areas. It is also a bridge to help solve problems into depth. In [28], Agarwal et al. first gave the following Cauchy–Schwarz inequality for Δ-integral on time scale.

Theorem A

Let\(\mathbb{T}\)be a time scale, \(t_{1},t_{2}\in\mathbb{T}\)with\(t_{1}< t_{2}\), and let\(s, t\in C_{rd}([t_{1}, t_{2}], \mathbb{R})\). Then
$$ \int_{t_{1}}^{t_{2}} \bigl\vert s(x)t(x) \bigr\vert \Delta x \leq \biggl( \int_{t_{1}}^{t_{2}}s^{2}(x)\Delta x \biggr)^{\frac{1}{2}} \biggl( \int _{t_{1}}^{t_{2}}t^{2}(x)\Delta x \biggr)^{\frac{1}{2}}. $$
(1)

Later, Wong et al. [29] presented the extension of inequality (1).

Theorem B

Let\(\mathbb{T}\)be a time scale, \(t_{1},t_{2}\in\mathbb{T}\)and\(t_{1}< t_{2}\), and let\(s, t, \lambda\in C_{rd}([t_{1}, t_{2}], \mathbb {R})\). Then
$$ \int_{t_{1}}^{t_{2}} \bigl\vert \lambda(x) \bigr\vert \bigl\vert s(x)t(x) \bigr\vert \Delta x \leq \biggl( \int_{t_{1}}^{t_{2}} \bigl\vert \lambda(x) \bigr\vert s^{2}(x)\Delta x \biggr)^{\frac {1}{2}} \biggl( \int_{t_{1}}^{t_{2}} \bigl\vert \lambda(x) \bigr\vert t^{2}(x)\Delta x \biggr)^{\frac{1}{2}}. $$

In 2008, Özkan et al. [30] introduced the time scale versions of (1) for the ∇-integral and \(\diamond_{\alpha }\)-integral, respectively.

Theorem C

Let\(\mathbb{T}\)be a time scale, \(t_{1},t_{2}\in\mathbb{T}\)and\(t_{1}< t_{2}\), and let\(s, t, \lambda\in C_{ld}([t_{1}, t_{2}], \mathbb {R})\). Then
$$ \int_{t_{1}}^{t_{2}} \bigl\vert \lambda(x) \bigr\vert \bigl\vert s(x)t(x) \bigr\vert \nabla x\leq \biggl( \int_{t_{1}}^{t_{2}} \bigl\vert \lambda(x) \bigr\vert s^{2}(x)\nabla x \biggr)^{\frac {1}{2}} \biggl( \int_{t_{1}}^{t_{2}} \bigl\vert \lambda(x) \bigr\vert t^{2}(x)\nabla x \biggr)^{\frac{1}{2}}. $$
(2)

Theorem D

Let\(\mathbb{T}\)be a time scale, \(t_{1},t_{2}\in\mathbb{T}\)with\(t_{1}< t_{2}\), and let\(s, t, \lambda: [t_{1}, t_{2}]\rightarrow\mathbb {R}\)be\(\diamond_{\alpha}\)-integrable functions. Then
$$ \int_{t_{1}}^{t_{2}} \bigl\vert \lambda(x) \bigr\vert \bigl\vert s(x)t(x) \bigr\vert \diamond_{\alpha} x\leq \biggl( \int_{t_{1}}^{t_{2}} \bigl\vert \lambda(x) \bigr\vert s^{2}(x)\diamond_{\alpha} x \biggr)^{\frac{1}{2}} \biggl( \int_{t_{1}}^{t_{2}} \bigl\vert \lambda(x) \bigr\vert t^{2}(x) \diamond _{\alpha}x \biggr)^{\frac{1}{2}} . $$
(3)

Remark 1.1

Taking \(\alpha=0\) in Theorem D, inequality (3) is reduced to inequality (2). Taking \(\alpha=1\) in Theorem D, inequality (3) is reduced to inequality (1).

In 2018, Tian [3] gave the triple diamond-alpha integral and proved that the Cauchy–Schwarz inequality holds for the triple diamond-alpha integral on the time scale.

Theorem E

Let\(\lambda(x_{1},x_{2},x_{3}), s(x_{1},x_{2},x_{3}), t(x_{1},x_{2},x_{3}): [a_{i},b_{i}]_{\mathbb {T}}^{3}\rightarrow\mathbb{R}\)be\(\diamond_{\alpha}\)-integrable functions with\(\lambda(x_{1},x_{2},x_{3})\geq0\). Then
$$ \begin{gathered} \biggl( \int_{a_{1}}^{b_{1}} \int_{a_{2}}^{b_{2}} \int_{a_{3}}^{b_{3}} \lambda(x_{1},x_{2},x_{3}) s(x_{1},x_{2},x_{3})t(x_{1},x_{2},x_{3}) \diamond_{\alpha} x_{1} \diamond_{\alpha} x_{2} \diamond_{\alpha} x_{3} \biggr)^{2} \\ \quad\leq \biggl( \int_{a_{1}}^{b_{1}} \int_{a_{2}}^{b_{2}} \int_{a_{3}}^{b_{3}} \lambda(x_{1},x_{2},x_{3})s^{2}(x_{1},x_{2},x_{3}) \diamond_{\alpha} x_{1} \diamond_{\alpha} x_{2} \diamond_{\alpha} x_{3} \biggr) \\ \qquad{}\times \biggl( \int_{a_{1}}^{b_{1}} \int_{a_{2}}^{b_{2}} \int_{a_{3}}^{b_{3}} \lambda(x_{1},x_{2},x_{3})t^{2}(x_{1},x_{2},x_{3}) \diamond_{\alpha} x_{1} \diamond_{\alpha} x_{2} \diamond_{\alpha} x_{3} \biggr) . \end{gathered} $$

In 2019, Tian et al. [4] introduced the notion of n-tuple diamond-alpha integral for a function of n variables and established the Cauchy–Schwarz inequality for n-tuple diamond-alpha integral as follows.

Theorem F

Let\(\lambda(\mathbf{x}), s(\mathbf{x}), t(\mathbf{x}): [a_{i},b_{i}]_{\mathbb{T}}^{n}\rightarrow \mathbb{R}\)be\(\diamond_{\alpha}\)-integrable functions with\(\lambda (\mathbf{x})\geq0\). Then
$$ \begin{aligned}[b] & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf{x}) s(\mathbf{x})t( \mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\leq \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})s^{2}( \mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})t^{2}( \mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr), \end{aligned} $$
(4)
where\(\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})\).

Moreover, Yeh et al. in [31] presented some interesting complements of the Cauchy–Schwarz inequality via delta integral. Motivated by the above results, in the paper, by using methods similar to that in [31], we shall give some new variants, generalizations, and refinements of the Cauchy–Schwarz inequality for n-tuple diamond-alpha integral on time scales.

2 Main results

Throughout the paper, we use \(\mathbb{T}\) to denote a time scale, denote \(\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})\), \([a_{i},b_{i}]^{n}_{\mathbb{T}}= [a_{1},b_{1}]\times[a_{2},b_{2}]\times\cdots\times[a_{n},b_{n}]\), where \(x_{i}, a_{i}, b_{i}\in\mathbb{T}\) with \(a_{i}< b_{i}\), \(i=1,2,\ldots,n\).

Proposition 2.1

([4])

Let\(s(\mathbf{x})\), \(t(\mathbf{x})\)be\(\diamond_{\alpha}\)-integrable functions on\([a_{i}, b_{i}]^{n}_{\mathbb{T}}\) (\(i=1,2,\ldots,n\)).
  1. (P1)
    If\(s(\mathbf{x})\geq0\)for\(\mathbf{x}\in[a_{i}, b_{i}]^{n}_{\mathbb {T}}\), then
    $$\int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}s(\mathbf{x})\diamond _{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n}\geq0. $$
     
  2. (P2)
    If\(s(\mathbf{x})\leq t(\mathbf{x})\)for\(\mathbf{x}\in[a_{i}, b_{i}]^{n}_{\mathbb{T}}\), then
    $$\begin{aligned} \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}s(\mathbf{x})\diamond _{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \leq \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}t(\mathbf{x})\diamond _{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n}. \end{aligned}$$
     
  3. (P3)
    If\(f(\mathbf{x})\geq0\)for\(\mathbf{x}\in[a_{i}, b_{i}]^{n}_{\mathbb {T}}\), then\(s(\mathbf{x})=0\)if and only if
    $$\int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}s(\mathbf{x})\diamond _{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n}=0. $$
     

Lemma 2.2

((AG inequality) [32])

Let\(r_{i}>0\) (\(i=1,2,\ldots,n\)), and let\(\theta_{1}, \theta_{2}, \ldots, \theta_{n}\in(1, +\infty)\)such that\(\sum_{i=1}^{n}\frac {1}{\theta_{i}}=1\). Then
$$ \prod_{i=1}^{n} r_{i} \leq\sum_{i=1}^{n} \frac{r_{i}^{\theta_{i}}}{\theta_{i}}. $$
(5)

Remark 2.3

The Cauchy–Schwarz inequality for n-tuple diamond-alpha integral has the following variants.
  1. (V1)
    Let \(t(\mathbf{x})>0\), \(p,q\in\mathbb{N}^{+}\), and let \(s(\mathbf {x})\) and \(t(\mathbf{x})\) be replaced by \(s^{p}(\mathbf{x})/\sqrt {t^{q}(\mathbf{x})}\) and \(\sqrt{t^{q}(\mathbf{x})}\) in (4), respectively. Then
    $$ \begin{aligned} & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s^{p}( \mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\leq \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \frac{h(\mathbf{x})s^{2p}(\mathbf{x})}{ t^{q}(\mathbf{x})} \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\qquad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})t^{q}( \mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr). \end{aligned} $$
     
  2. (V2)
    Let \(s(\mathbf{x})\) and \(t(\mathbf{x})\) be replaced by \((s^{2p}(\mathbf {x})+t^{2q}(\mathbf{x}))^{\frac{1}{2}}\) and \(s^{p}(\mathbf{\mathbf {x}})t^{q}(\mathbf{\mathbf{x}})/(s^{2p}(\mathbf{x}) +t^{2q}(\mathbf{x}))^{\frac{1}{2}}\) in (4), respectively, where \(p,q\in\mathbb{N}^{+}\). Then we get
    $$ \begin{aligned} & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s^{p}( \mathbf{x})t^{q}(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\leq \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x}) \bigl(s^{2p}(\mathbf{x})+t^{2q}(\mathbf{x})\bigr) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\qquad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\frac{ \lambda(\mathbf{x}) s^{2p}(\mathbf{x})t^{2q}(\mathbf{x})}{ s^{2p}(\mathbf{x})+t^{2q}(\mathbf{x})} \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) . \end{aligned} $$
     
  3. (V3)
    Let \(s(\mathbf{x})\) and \(t(\mathbf{x})\) be replaced by \(\sqrt {t^{q}(\mathbf{x})/s^{p}(\mathbf{x})}\) and \(\sqrt{s^{p}(\mathbf{x})t^{q}(\mathbf {x})}\) in (4), respectively, where \(s^{p}(\mathbf{x})t^{q}(\mathbf {x})\geq0\), \(s(\mathbf{x})\neq0\), and \(p,q\in\mathbb{N}^{+}\). Then we get
    $$ \begin{aligned} & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})t^{q}( \mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\leq \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \frac{\lambda(\mathbf{x})t^{q}(\mathbf{x})}{ s^{p}(\mathbf{x})} \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\qquad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s^{p}( \mathbf{x})t^{q}(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n} \biggr). \end{aligned} $$
     

Theorem 2.4

Let\(\lambda(\mathbf{x}), s(\mathbf{x}), t(\mathbf{x}):[a_{i},b_{i}]_{\mathbb{T}}^{n}\rightarrow \mathbb{R}\)be\(\diamond_{\alpha}\)-integrable functions with\({\lambda (\mathbf{x})}\geq0\), and let there be constants\(m,M, h, H\in\mathbb {R}\)such that
$$ \bigl(Ms(\mathbf{x})-ht(\mathbf{x})\bigr) \bigl(Ht(\mathbf{x})-ms( \mathbf{x})\bigr)\geq0. $$
(6)
Then
$$ \begin{aligned}[b]&mM \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\quad\quad +hH \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\quad\leq(mh+MH) \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \\ &\quad\leq \vert mh+MH \vert \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda (\mathbf{x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{1/2} \\ & \quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}}\lambda(\mathbf{x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{1/2}.\end{aligned} $$
(7)
Specially, if\(Mm>0\), \(Hh>0\), then
$$ \begin{aligned}[b]& \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\leq\frac{ (hm+HM )^{2}}{4hmHM} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n} \biggr)^{2}.\end{aligned} $$
(8)

Proof

It is easy to find from (6) that
$$ \lambda(\mathbf{x}) \bigl(Ms(\mathbf{x})-ht(\mathbf{x})\bigr) \bigl(Ht(\mathbf{x})-ms( \mathbf {x})\bigr)\geq0. $$
Then
$$\begin{aligned} & \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} MH\lambda(\mathbf {x})t(\mathbf{x})s( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \\ &\quad- \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} hH\lambda(\mathbf{x}) t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ & \quad-mM \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\quad+mh \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x})s( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n}\geq0. \end{aligned}$$
(9)
From (9) and Cauchy–Schwarz inequality (4), we find that (7) holds.
Moreover, from \(mM>0\), \(hH>0\), and
$$\begin{aligned} & \biggl[ \biggl(hH \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \biggr)^{\frac{1}{2}} \\ &\quad - \biggl(mM \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{\frac{1}{2}} \biggr]^{2}\geq0, \end{aligned}$$
(10)
we have
$$\begin{aligned} &hH \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x}) t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\qquad+ mM \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ & \quad\geq2 \biggl(Mm \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{1/2} \\ &\quad\quad\times \biggl(Hh \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{1/2}. \end{aligned}$$
(11)

Therefore, by using (11) and (7), we find that (8) is valid. The proof of Theorem 2.10 is completed. □

Remark 2.5

Obviously, inequality (8) extends the result in [33].

Remark 2.6

Under the assumptions of Theorem 2.4, and letting \(mM>0\), \(hH>0\), \(0< q\leq p<1\), and \(p+q=1\), from the AG inequality (5) and (7) we have
$$\begin{aligned} & \biggl(\frac{mM}{p} \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}}\lambda(\mathbf{x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{p} \\ &\quad\quad{} \cdot \biggl(\frac{hH}{q} \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}}\lambda(\mathbf{x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{q} \\ &\quad\leq mM \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\quad\quad{} +hH \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\quad\leq(mh+MH) \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n}, \end{aligned}$$
which implies that
$$\begin{aligned} &\biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{q} \\ &\qquad{}\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{p} \\ &\quad\leq p^{p}(1-p)^{1-p}\frac{mh+MH}{(mM)^{p}(hH)^{1-p}} \int _{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf{x})s(\mathbf {x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n}. \end{aligned}$$
(12)
Letting \(p\rightarrow1^{-}\) on the both sides of inequality (12), we find
$$\begin{aligned} & \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\quad\leq \biggl(\frac{H}{m}+\frac{h}{M} \biggr) \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}}\lambda(\mathbf{x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha } x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\quad= \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})s(\mathbf{x}) \frac{Ht(\mathbf{x})}{m}\diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n} \\ &\quad\quad {}+ \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf{x})s(\mathbf{x}) \frac{ht(\mathbf{x})}{M}\diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n}. \end{aligned}$$
Letting \(p\rightarrow0^{+}\) on the both sides of inequality (12), we find
$$\begin{aligned} & \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\quad \leq \biggl(\frac{m}{H}+\frac{M}{h} \biggr) \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}} \lambda(\mathbf{x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n}. \end{aligned}$$
(13)

Remark 2.7

Let \(S(\mathbf{x}):[a_{i},b_{i}]_{\mathbb{T}}^{n}\rightarrow(0,+\infty)\) be a \(\diamond_{\alpha}\)-integrable function. If \(s(\mathbf{x}) = S^{\frac {-1}{2}}(\mathbf{x})\), \(t(\mathbf{x})= S^{\frac{1}{2}}(\mathbf{x})\), then inequality (7) reduces to
$$\begin{aligned} &hH \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})S(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} +mM \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \frac{\lambda(\mathbf{x})}{S(\mathbf{x})} \diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n} \\ & \quad\leq(mh+MH) \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n}, \end{aligned}$$
which generalizes the result in [34].

Remark 2.8

Letting \(\lambda(\mathbf{x})=1\) in (8), then inequality (8) reduces to
$$\begin{aligned} & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}s^{2}(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}t^{2}(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\leq\frac{ (hm+HM )^{2}}{4hmHM} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} s(\mathbf{x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2}, \end{aligned}$$
which extends Pólya and Szegö’s result [35].

Remark 2.9

Let \(S(\mathbf{x}):[a_{i},b_{i}]_{\mathbb{T}}^{n}\rightarrow(0,+\infty)\) be a \(\diamond_{\alpha}\)-integrable function. If \(s(\mathbf{x})= S^{\frac {-1}{2}}(\mathbf{x})\), \(t(\mathbf{x})= S^{\frac{1}{2}}(\mathbf{x})\), then inequality (8) reduces to
$$\begin{aligned} & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \frac{\lambda(\mathbf{x})}{S(\mathbf{x})} \diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})S(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr) \\ &\quad\leq\frac{ (hm+HM )^{2}}{4hmHM} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2}, \end{aligned}$$
which extends some results in [36, 37].

Remark 2.10

Letting \(h=H=1\) in (8), inequality (8) reduces to
$$ \begin{aligned}[b] & \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x}) t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\quad\quad +mM \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \\ &\quad\leq(m+M) \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x})s( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \\ &\quad\leq \vert m+M \vert \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \biggr)^{\frac{1}{2}} \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr)^{\frac{1}{2}}. \end{aligned} $$
(14)
Moreover, if \(mM>0\), then
$$\begin{aligned} & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad{}\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\leq\frac{ (m+M )^{2}}{4mM} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}} \lambda(\mathbf{x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n} \biggr)^{2}. \end{aligned}$$
(15)
The above inequalities (14) and (15) extend some results in [31].

Theorem 2.11

Let\(\lambda(\mathbf{x})\), \(s(\mathbf{x})\), \(t(\mathbf{x}):[a_{i},b_{i}]_{\mathbb{T}}^{n}\rightarrow [0, +\infty)\)be\(\diamond_{\alpha}\)-integrable functions, let there be constants\(h, H, m, M, p, q>0\)such that\(m\leq t(\mathbf{x})\leq M\), \(h\leq s(\mathbf{x})\leq H\), \(0< q\leq p<1\), and\(p+q=1\). Then
$$\begin{aligned} & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n} }\lambda(\mathbf{x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr)^{p} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf{x}) t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{q} \\ &\quad \leq\frac{p HM+q hm}{(hH)^{q}(mM)^{p}} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}} \lambda(\mathbf{x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n} \biggr). \end{aligned}$$
(16)

Proof

As \((pMs(\mathbf{x})-hqt(\mathbf{x}))(ms(\mathbf{x})-Ht(\mathbf{x}))\leq0\), we have
$$ pmMs^{2}(\mathbf{x})-(p HM+q hm)s(\mathbf{x})t(\mathbf{x})+q Hht^{2}(\mathbf {x})\leq0. $$
Then
$$ pmMs^{2}(\mathbf{x})+q Hht^{2}(\mathbf{x})\leq(p HM+q hm)s(\mathbf{x})t(\mathbf{x}). $$
(17)
By the AG inequality (5) and (17), we find
$$\begin{aligned} & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n} }\lambda(\mathbf{x})s^{2}( \mathbf{x}) \biggr)^{p} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}}\lambda(\mathbf{x}) t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{q} \\ &\quad=\frac{1}{(hH)^{q}(mM)^{p}} \biggl(mM \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr)^{p} \\ &\quad\quad\times \biggl(hH \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr)^{q} \\ &\quad\leq\frac{1}{(hH)^{q}(mM)^{p}} \biggl(pmM \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}} \lambda(\mathbf{x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \\ &\quad\quad +q hH \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr) \\ &\quad\leq\frac{p HM+q hm}{(hH)^{q}(mM)^{p}} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots \diamond_{\alpha} x_{n} \biggr), \end{aligned}$$
which implies (16) holds. □

Theorem 2.12

Let\(\lambda(\mathbf{x}), s(\mathbf{x}), t(\mathbf{x}):[a_{i},b_{i}]_{\mathbb{T}}^{n}\rightarrow [0, +\infty)\)be\(\diamond_{\alpha}\)-integrable functions.
  1. (i)
    If there are constants\(h,H,m,M\in\mathbb{R}\)such that\([Ht(\mathbf {x})-ms(\mathbf{y})][Ms(\mathbf{y})-ht(\mathbf{x})]\geq0\)for all\(\mathbf{x}, \mathbf{y}\in[a_{i},b_{i}]_{\mathbb{T}}^{n}\), then
    $$\begin{aligned} & (hm+HM ) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\geq hH \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad +mM \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr). \end{aligned}$$
    (18)
     
  2. (ii)
    If\(hH>0\), \(mM>0\)such that\([Ht(\mathbf{x})-ms(\mathbf{y})][Ms(\mathbf {y})-ht(\mathbf{x})]\geq0\)for all\(\mathbf{x}, \mathbf{y}\in [a_{i},b_{i}]_{\mathbb{T}}^{n}\), and if\(p, q>0\)such that\(\frac {1}{p}+\frac{1}{q}=1\), then
    $$\begin{aligned} & (hm+HM ) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\geq \biggl[ \frac{hH}{p} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggr]^{p} \\ &\quad\quad\times \biggl[ \frac{mM}{q} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggr]^{q} . \end{aligned}$$
    (19)
     
  3. (iii)
    If\(mM>0\), \(hH>0\)such that\([Ht(\mathbf{x})-ms(\mathbf{x}) ] [Ms(\mathbf{x})-ht(\mathbf{x}) ]\geq0\), then
    $$\begin{aligned} &(hm+HM) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad{}\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\geq hH \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad{} +mM \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2}. \end{aligned}$$
     
  4. (iv)
    If\(mM>0\), \(hH>0\)such that\([Ht(\mathbf{x})-ms(\mathbf{y})][Ms(\mathbf {y})-ht(\mathbf{x})]\geq0\)for all\(\mathbf{x}, \mathbf{y}\in [a_{i},b_{i}]_{\mathbb{T}}^{n}\), then
    $$\begin{aligned} & (hm+HM) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad{}\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \biggr) \\ &\quad\geq hH \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad{} +mM \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2}. \end{aligned}$$
     

Proof

Case (i). From the assumption we find that
$$ \lambda(\mathbf{x})\lambda(\mathbf{y}) \bigl(Ht(\mathbf{x})-m s(\mathbf{y}) \bigr) \bigl(Ms(\mathbf{y})-ht(\mathbf{x}) \bigr)\geq0, $$
which means that
$$\begin{aligned} &HM\lambda(\mathbf{x})\lambda(\mathbf{y})s(\mathbf{y})t(\mathbf{x}) +hm\lambda(\mathbf{x}) \lambda(\mathbf{y})s(\mathbf{y})t(\mathbf{x}) \\ &\quad\geq hH\lambda(\mathbf{x})\lambda(\mathbf{y})t^{2}(\mathbf{x}) +mM\lambda( \mathbf{x})\lambda(\mathbf{y})s^{2}(\mathbf{x}). \end{aligned}$$
Therefore
$$\begin{aligned} & (hm+HM ) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {y})s(\mathbf{y}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\geq hH \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad +mM \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr). \end{aligned}$$

Case (ii). From AG inequality (5) and Case (i), it is easy to find that (19) holds.

Case (iii). From Cauchy–Schwarz inequality (4) we find that
$$\begin{aligned} & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad \geq \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2}, \end{aligned}$$
(20)
and
$$\begin{aligned} & \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ & \quad\geq \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2}. \end{aligned}$$
(21)
Combining (7), (20), and (21), we have
$$\begin{aligned} & (hm+HM) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \biggr) \\ &\quad\geq hH \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad +mM \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\geq hH \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad +mM \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2}. \end{aligned}$$
The proof of Case (iii) is completed.
Case (iv). Combining (18), (20), and (21), we have
$$\begin{aligned} & (hm+HM) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\geq hH \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad +mM \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\geq hH \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ & \quad\quad +mM \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2}, \end{aligned}$$
which implies that Case (iv) holds. □

Remark 2.13

From Case (i) of Theorem 2.12 we find that
$$\begin{aligned} & (hm+HM )^{2} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}} \lambda(\mathbf{x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\geq h^{2}H^{2} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad +m^{2}M^{2} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\qquad+ 2hmHM \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\geq 4hmHM \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} . \end{aligned}$$
Therefore, if \(hmHM>0\), we find
$$\begin{aligned} & (hm+HM ) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\geq 2\sqrt{hmHM} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) . \end{aligned}$$
Similarly, from Case (iv) of Theorem 2.12 we have
$$\begin{aligned} & (hm+HM) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad{}\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \biggr) \\ &\quad\geq 2\sqrt{hmHM} \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad{}\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr). \end{aligned}$$

By using methods similar to that in [31], we can prove the following theorem.

Theorem 2.14

Let\(\lambda(\mathbf{x}), s(\mathbf{x}), t(\mathbf{x}):[a_{i},b_{i}]_{\mathbb{T}}^{n}\rightarrow [0,+\infty)\)be\(\diamond_{\alpha}\)-integrable functions.

Case (1).
$$\begin{aligned} & \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \biggr] \\ & \qquad\times \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \biggr] \\ &\quad\geq \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x})s(\mathbf{x})t( \mathbf{x}) \biggr) \\ &\quad\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}} \lambda(\mathbf{x})t(\mathbf{x}) \biggr) \biggr]^{2}. \end{aligned}$$
Case (2). If there are constants\(h,H,m,M\in\mathbb{R}\)such that\([Ht(\mathbf{x})-ms(\mathbf{x}) ] [Ms(\mathbf{x})-ht(\mathbf{x}) ]\geq 0\), then
$$\begin{aligned} & (hm+HM )^{2} \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int _{a_{n}}^{b_{n}} \lambda(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond _{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \biggr) \\ &\quad\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggr]^{2} \\ &\quad\geq 4hmHM \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \biggr] \\ &\quad \quad\times \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)\\ &\qquad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda (\mathbf{x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \biggr]^{2}. \end{aligned}$$
Case (3). If there are constants\(h,H,m,M\in\mathbb{R}\)such that\([Ht(\mathbf{x})-ms(\mathbf{y})][Ms(\mathbf{y})-ht(\mathbf{x})]\geq0\)for all\(\mathbf{x}, \mathbf{y}\in[a_{i},b_{i}]_{\mathbb{T}}^{n}\), then
$$\begin{aligned} 1& \leq \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \biggr] \\ &\quad\times \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \biggr] \\ &\quad \Big/ \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \biggr) \\ &\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggr]^{2} \\ &\leq \frac{ (hm+HM )^{2}}{4hmHM}. \end{aligned}$$

Notes

Acknowledgements

The authors gratefully acknowledge the anonymous referees for their constructive comments and advices on the earlier version for this paper.

Availability of data and materials

Not applicable.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Funding

This work was supported by the Fundamental Research Funds for the Central Universities (No. 2015ZD29).

Competing interests

The authors declare that they have no competing interests.

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Authors and Affiliations

  1. 1.School of Economics and ManagementNorth China Electric Power UniversityBaodingP.R. China
  2. 2.Department of Mathematics and PhysicsNorth China Electric Power UniversityBaodingP.R. China
  3. 3.Department of MathematicsHuzhou UniversityHuzhouP.R. China

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