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 [2–12] 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 [13–27]). 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

$$\begin{aligned} 1& \leq \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \biggr] \\ &\quad\times \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t^{2}( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr)^{2} \biggr] \\ &\quad \Big/ \biggl[ \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x})t( \mathbf{x})\diamond_{\alpha} x_{1}\cdots\diamond_{\alpha } x_{n} \biggr) \\ &\quad + \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})s(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \\ &\quad\times \biggl( \int_{a_{1}}^{b_{1}}\cdots \int_{a_{n}}^{b_{n}} \lambda(\mathbf {x})t(\mathbf{x}) \diamond_{\alpha} x_{1}\cdots\diamond_{\alpha} x_{n} \biggr) \biggr]^{2} \\ &\leq \frac{ (hm+HM )^{2}}{4hmHM}. \end{aligned}$$