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Some dynamic Hilbert-type inequalities for two variables on time scales

Abstract

In this paper, we discuss some new Hilbert-type dynamic inequalities on time scales in two separate variables. We also deduce special cases, like some integral and their respective discrete inequalities.

Introduction

It is evident that Hilbert-type inequalities play a major role in mathematics, for complex pattern analysis, numerical analysis, qualitative theory of differential equations and their implementations. Hilbert’s discrete inequality and its integral formula [1, Theorem 316] have been generalized in many ways (for example, see [26]). Lately, Pachpatte in [6], obtained the following inequality: if \(A_{q}=\sum_{s=1}^{q}a_{s}\geq 0\) and \(B_{n}=\sum_{t=1}^{n}b_{t}\geq 0\), for \(q=1,2,\dots ,p\) and \(n=1, 2,\dots ,r\), where p and r are the natural numbers, then

$$\begin{aligned} \sum_{q=1}^{p}\sum _{n=1}^{r}\frac{A_{q}B_{n}}{q+n} \leq &C(p, r) \Biggl( \sum_{q=1}^{p}(p-q+1) (a_{q})^{2} \Biggr) ^{\frac{1}{2}} \\ &{}\times \Biggl( \sum_{n=1}^{r}(r-n+1) (b_{n})^{2} \Biggr) ^{ \frac{1}{2}}, \end{aligned}$$
(1)

where

$$ C(p, r)=\frac{1}{2}\sqrt{pr}. $$

The integral analogue of (1) is given by

$$\begin{aligned} \int _{0}^{x} \int _{0}^{y}\frac{F(s)G(t)}{s+t}\,ds \,dt \leq &D(x, y) \biggl( \int _{0}^{x}(x-s)f^{2}(s)\,ds \biggr) ^{\frac{1}{2}} \\ &{}\times \biggl( \int _{0}^{y}(y-t)g^{2}(t)\,dt \biggr) ^{\frac{1}{2}}, \end{aligned}$$
(2)

where \(F(s)=\int _{0}^{s}f(\tau )\,d\tau \geq 0\), \(G(t)=\int _{0}^{t}g(\nu )\,d\nu \geq 0\), and

$$ D(x, y)=\frac{1}{2}\sqrt{xy}. $$

In the past few years, several researchers have suggested the study of dynamic time scale inequalities. In [7] the authors deduced some generalizations of the inequalities (1) and (2) on time scales. Namely, they proved that if \(A(x_{1})=\int _{w}^{x_{1}}a(\tau _{1})\Delta \tau _{1}\), \(B(y_{1})=\int _{w}^{y_{1}}b(\tau _{1})\Delta \tau _{1}\), and \(p_{1}>1\), \(q_{1}>1\) with \(p_{1}^{-1}+q_{1}^{-1}=1\), then

$$\begin{aligned} & \int _{w}^{x_{2}} \int _{w}^{y_{2}} \frac{A(x_{1})B(y_{1})}{q_{1}(x_{1}-w)^{p_{1}-1}+p_{1}(y_{1}-w)^{q_{1}-1}}\Delta y_{1}\Delta x_{1} \\ &\quad \leq M(p_{1}, q_{1}) \biggl( \int _{w}^{x_{2}}\bigl(\sigma (x_{2})-x_{1} \bigr) \bigl(a(x_{1})\bigr)^{p_{1}} \Delta x_{1} \biggr) ^{p_{1}^{-1}} \\ &\qquad {}\times ( \int _{w}^{y_{2}}\bigl(\sigma (y_{2})-y_{1} \bigr) \bigl(b(y_{1})^{q_{1}} \Delta y_{1} \bigr) ^{q_{1}^{-1}}, \end{aligned}$$
(3)

where

$$ M(p_{1}, q_{1})=(p_{1}q_{1})^{-1}(x_{2}-w)^{p_{1}^{-1}(p_{1}-1)}(y_{2}-w)^{q_{1}^{-1}(q_{1}-1)}. $$
(4)

In order to develop dynamic time scale inequalities, we moved the reader to the articles [819].

Motivated by the above results, our major aim in this paper is to establish some dynamic Hilbert-type inequalities in two separate variables on time scales. These inequalities can be considered as extensions and generalizations of some Hilbert-type inequalities proved in [7] for the two-dimensional on time scales.

The paper is governed as follows: In Sect. 2, we remember some basic notions, definitions and results on time scales calculus which will be required in proving our main outcomes. In Sect. 3, we will exemplify the major results.

Preliminaries and basic lemmas

In this section, we will present some fundamental concepts and effects on time scales which will be beneficial for deducing our main results. The following definitions and theorems are referred from [20, 21].

A time scale \(\mathbb{T}\) is defined as an arbitrary nonempty closed subset of the real numbers. We define the forward jump operator \(\sigma :\mathbb{T}\rightarrow \mathbb{T}\) for any \(t^{\ast }\in \mathbb{T}\) by

$$ \sigma \bigl(t^{\ast }\bigr)=\inf \bigl\{ s^{\ast }\in \mathbb{T}:s^{\ast }>t^{\ast } \bigr\} , $$

and the backward jump operator \(\rho :\mathbb{T}\rightarrow \mathbb{T}\) for any \(t^{\ast }\in \mathbb{T}\) by

$$ \rho \bigl(t^{\ast }\bigr)=\sup \bigl\{ s^{\ast }\in \mathbb{T}:s^{\ast }< t^{\ast }\bigr\} . $$

From the above two definitions, it can be stated that a point \(t^{\ast }\in \mathbb{T}\) with \(\inf \mathbb{T}< t^{\ast }<\sup \mathbb{T}\) is called right-scattered if \(\sigma (t^{\ast })>t^{\ast }\), right-dense if \(\sigma (t^{\ast })=t^{\ast }\), left-scattered if \(\rho (t^{\ast })< t^{\ast }\), and left-dense if \(\rho (t^{\ast })=t^{\ast }\). If \(\mathbb{T}\) has a left-scattered maximum \(t_{m}^{\ast }\), then \(\mathbb{T}^{k}=\mathbb{T}-\{t_{m}^{\ast }\}\), otherwise \(\mathbb{T}^{k}=\mathbb{T}\). Moreover, the forward graininess function \(\mu :\mathbb{T}\rightarrow {[} 0,\infty )\) for any \(t^{\ast }\in \mathbb{T}\) is defined by \(\mu (t^{\ast })=\sigma (t^{\ast })-t^{\ast }\).

For a function \(f:\mathbb{T}\rightarrow \mathbb{R}\), the delta derivative of f at \(t^{\ast }\in \mathbb{T}^{k}\) is defined as \(f^{\Delta }(t^{\ast }) \) if for each \(\varepsilon >0\) there exists a neighborhood \(U^{\ast }\) of \(t^{\ast }\) such that

$$ \bigl\vert {}\bigl[ f\bigl(\sigma \bigl(t^{\ast }\bigr)\bigr)-f \bigl(s^{\ast }\bigr)\bigr]-f^{\Delta }\bigl(t^{ \ast }\bigr) \bigl[\sigma \bigl(t^{\ast }\bigr)-s^{\ast }\bigr] \bigr\vert \leq \varepsilon \bigl\vert \sigma \bigl(t^{\ast }\bigr)-s^{\ast } \bigr\vert , \quad \text{for all }s^{\ast }\in U^{\ast }. $$

A function \(f:\mathbb{T}\rightarrow \mathbb{R}\) is called right-dense continuous (rd-continuous) if it is continuous at all right-dense points in \(\mathbb{T}\) and its left-sided limits exist (finite) at all left-dense points in \(\mathbb{T}\). The set of all such rd-continuous functions is denoted by \(\mathrm{C}_{rd}(\mathbb{T}, \mathbb{R})\). We will frequently use the following useful relations between calculus on time scales \(\mathbb{T}\) and differential calculus on \(\mathbb{R}\), as well as difference calculus on \(\mathbb{Z}\). Note that

(i) if \(\mathbb{T}=\mathbb{R}\), then

$$ \begin{gathered} \sigma \bigl(t^{\ast }\bigr)=t^{\ast },\qquad \mu \bigl(t^{\ast } \bigr)=0,\qquad f^{\Delta }\bigl(t^{\ast }\bigr)=f^{\prime } \bigl(t^{\ast }\bigr), \quad \text{and}\\ \int _{a}^{b}f^{\Delta }\bigl(t^{\ast } \bigr)\Delta t^{\ast }= \int _{a}^{b}f\bigl(t^{ \ast }\bigr) \,dt^{\ast }; \end{gathered} $$
(5)

(ii) if \(\mathbb{T}=\mathbb{Z}\), then

$$ \begin{gathered} \sigma \bigl(t^{\ast }\bigr)=t^{\ast }+1, \qquad \mu \bigl(t^{\ast }\bigr)=1,\qquad f^{\Delta }\bigl(t^{\ast }\bigr)= \Delta f\bigl(t^{\ast }\bigr), \quad \text{and}\\ \int _{a}^{b}f^{\Delta }\bigl(t^{\ast } \bigr)\Delta t^{\ast }=\sum_{t^{\ast }=a}^{b-1}f \bigl(t^{ \ast }\bigr).\end{gathered} $$
(6)

Also, we must know some essentials about partial derivatives on time scales. Let \(\mathbb{T}_{1}\) and \(\mathbb{T}_{2}\) be any two time scales. Let \(\sigma _{1}\), \(\Delta _{1}\) and \(\sigma _{2}\), \(\Delta _{2}\) denote the forward jump operator and the delta differentiation operator on \(\mathbb{T}_{1}\) and \(\mathbb{T}_{2}\), respectively. Assume that \(u< w\) are points in \(\mathbb{T}_{1}\), \(e< f\) are points in \(\mathbb{T}_{2}\), \([u, w)\) is a semiclosed bounded interval in \(\mathbb{T}_{1}\), and \([e, f)\) is a semiclosed bounded interval in \(\mathbb{T}_{2}\). Let us consider a “rectangle” in \(\mathbb{T}_{1}\times \mathbb{T}_{2}\) given by

$$ \mathrm{R}=[u, w)_{\mathbb{T}_{1}}\times {[} e, f)_{\mathbb{T}_{2}}=\bigl\{ \bigl(t_{{1}}^{\ast }, t_{{2}}^{\ast } \bigr):t_{{1}}^{ \ast }\in {[} u, v)_{\mathbb{T}_{1}}, t_{{2}}^{\ast } \in {[} e, f)_{\mathbb{T}_{2}}\bigr\} . $$

Suppose \(f:\mathbb{T}_{1}\times \mathbb{T}_{2}\rightarrow \mathbb{R}\) is a real-valued function. At \((t_{{1}}^{\ast }, t_{{2}}^{\ast })\in \mathbb{T}_{1}\times \mathbb{T}_{2}\), we say that f has a \(\Delta _{1}\) partial derivative with respect to \(t_{{1}}^{\ast }\) if for each \(\varepsilon >0\) there exists a neighborhood \(U_{t_{{1}}^{\ast }}\) of \(t_{{1}}^{\ast }\) such that

$$ \bigl\vert {}\bigl[ f\bigl(\sigma _{1}\bigl(t_{{1}}^{\ast } \bigr), t_{{2}}^{ \ast }\bigr)-f\bigl(s^{\ast }, t_{{2}}^{\ast }\bigr)\bigr]-f^{\Delta _{1}}\bigl(t_{{1}}^{ \ast },t_{{2}}^{\ast } \bigr)\bigl[\sigma _{1}\bigl(t_{{1}}^{\ast } \bigr)-s^{\ast }\bigr] \bigr\vert \leq \varepsilon \bigl\vert \sigma _{{_{1}}}\bigl(t_{{1}}^{ \ast }\bigr)-s^{\ast } \bigr\vert , $$

for all \(s^{\ast }\in U_{t_{{1}}^{\ast }}\). At \((t_{{1}}^{\ast }, t_{{2}}^{\ast })\in \mathbb{T}_{1}\times \mathbb{T}_{2}\), we say that f has a \(\Delta _{2}\) partial derivative with respect to \(t_{{2}}^{\ast }\) if for each \(\varepsilon >0\) there exists a neighborhood \(U_{t_{{2}}^{\ast }}\) of \(t_{{2}}^{\ast }\) such that

$$ \bigl\vert {}\bigl[ f\bigl(t_{{1}}^{\ast }, \sigma _{{2}}\bigl(t_{{2}}^{ \ast }\bigr)\bigr)-f \bigl(t_{{1}}^{\ast }, l^{\ast }\bigr)\bigr]-f^{\Delta _{2}} \bigl(t_{{1}}^{ \ast },t_{2}^{\ast }\bigr)\bigl[ \sigma _{2}\bigl(t_{2}^{\ast }\bigr)-l^{\ast } \bigr] \bigr\vert \leq \varepsilon \bigl\vert \sigma _{{_{2}}} \bigl(t_{{2}}^{\ast }\bigr)-l^{ \ast } \bigr\vert , $$

for all \(l^{\ast }\in U_{t_{{2}}^{\ast }}\).

A function \(f:\mathbb{T}_{1}\times \mathbb{T}_{2}\rightarrow \mathbb{R}\) is said to be rd-continuous in \(t_{2}^{\ast }\) if for every \(\beta _{1}^{\ast }\in \mathbb{T}_{1}\), the function \(f(\beta _{1}^{\ast }, t_{2}^{\ast })\) is rd-continuous on \(\mathbb{T}_{2}\) and is rd-continuous in \(t_{{1}}^{\ast }\) if for every \(\beta _{2}^{\ast }\in \mathbb{T}_{2}\), the function \(f(t_{{1}}^{\ast }, \beta _{2}^{\ast })\) is rd-continuous on \(\mathbb{T}_{1}\). Let \(\mathrm{CC}_{rd}\) denote the set of functions \(f(t_{{1}}^{\ast }, t_{{2}}^{\ast })\) on \(\mathbb{T}_{1}\times \mathbb{T}_{2}\) with the properties:

(A1) f is rd-continuous in \(t_{1}^{\ast }\);

(A2) f is rd-continuous in \(t_{2}^{\ast }\);

(A3) if \((x_{1}^{\ast }, x_{2}^{\ast })\in \mathbb{T}_{1}\times \mathbb{T}_{2}\) with \(x_{1}^{\ast }\) right-dense or maximal and \(x_{2}^{\ast }\) right-dense or maximal, then f is continuous at \((x_{1}^{\ast }, x_{2}^{\ast })\);

(A4) if \(x_{1}^{\ast }\) and \(x_{1}^{\ast }\) are both left-dense, then the limit of \(f(t_{1}^{\ast }, t_{2}^{\ast })\) exists as \((t_{1}^{\ast }, t_{2}^{\ast })\) approaches \((x_{1}^{\ast }, x_{2}^{\ast })\) along any path in the region

$$ \mathrm{R}_{LL}\bigl(x_{1}^{\ast }, x_{2}^{\ast }\bigr)=\bigl\{ \bigl(t_{1}^{\ast }, t_{2}^{\ast }\bigr):t_{1}^{\ast }\in {[} u, x_{1}^{\ast }) \cap \mathbb{T}_{1}, t_{2}^{\ast }\in {[} e, x_{2}^{ \ast })\cap \mathbb{T}_{2}\bigr\} . $$

Let \(\mathrm{CC}_{rd}^{1}\) be the set of all functions in \(\mathrm{CC}_{rd}\) for which both the \(\Delta _{1}\) partial derivative and the \(\Delta _{2}\) partial derivative exist in \(\mathrm{CC}_{rd}\).

In the following, we present Fubini theorem on time scales which plays a key role in proving the main results of this paper.

Theorem 1

(Fubini’s theorem [22])

Let \(\mathbb{T}_{1}\) and \(\mathbb{T}_{2}\) be two time scales. Suppose that \(f:\mathbb{T}_{1}\times \mathbb{T}_{2}\rightarrow \mathbb{R}\) is a Δ-integrable function with respect to both time scales. Define

$$ \phi \bigl(t^{\ast }\bigr)= \int _{\mathbb{T}_{1}}f\bigl(x^{\ast }, t^{\ast }\bigr) \Delta x^{\ast },\quad \textit{for a.e. }t^{\ast }\in \mathbb{T}_{1}, $$

and

$$ \psi \bigl(x^{\ast }\bigr)= \int _{\mathbb{T}_{2}}f\bigl(x^{\ast }, t^{\ast }\bigr) \Delta t^{\ast },\quad \textit{for a.e. }x^{\ast }\in \mathbb{T}_{2}. $$

Then ϕ and ψ are Δ-integrable on \(\mathbb{T}_{1}\), \(\mathbb{T}_{2}\), respectively, and

$$ \int _{\mathbb{T}_{1}}\Delta x^{\ast } \int _{\mathbb{T}_{2}}f\bigl(x^{ \ast }, t^{\ast }\bigr)\Delta t^{\ast }= \int _{\mathbb{T}_{2}} \Delta t^{\ast } \int _{\mathbb{T}_{1}}f\bigl(x^{\ast }, t^{\ast }\bigr)\Delta x^{\ast }. $$
(7)

Next, we present Hölder’s and Jensen’s inequalities in two dimensions on time scales.

Theorem 2

(Hölder’s inequality [23, Theorem 2.3.10])

Let \(u, v \in \mathbb{T}\) with \(u< v\). If \(f, g\in \mathrm{CC}_{rd}^{1}([u, v]_{\mathbb{T}}\times {[} u, v]_{\mathbb{T}}, \mathbb{R})\) are integrable functions and \(p^{-1}+q^{-1}=1\) with \(p>1\), then

$$\begin{aligned} \int _{u}^{v} \int _{u}^{v} \bigl\vert f\bigl(r^{\ast }, t^{\ast }\bigr)g\bigl(r^{\ast }, t^{\ast }\bigr) \bigr\vert \Delta r^{\ast }\Delta t^{\ast } \leq & \biggl( \int _{u}^{v} \int _{u}^{v} \bigl\vert f\bigl(r^{\ast }, t^{\ast }\bigr) \bigr\vert ^{p}\Delta r^{\ast } \Delta t^{\ast } \biggr) ^{p^{-1}} \\ &{}\times \biggl( \int _{u}^{v} \int _{u}^{v} \bigl\vert g\bigl(r^{\ast }, t^{ \ast }\bigr) \bigr\vert ^{q}\Delta r^{\ast }\Delta t^{\ast } \biggr) ^{q^{-1}}. \end{aligned}$$
(8)

Theorem 3

(Jensen’s inequality [24, Theorem 3.1])

Let \(r^{\ast }, t^{\ast }\in \mathrm{R}\) and \(-\infty \leq m^{\ast }< n^{\ast }\leq \infty \). If \(f\in \mathrm{CC}_{rd}^{1}(\mathbb{R}, (m^{\ast }, n^{\ast }))\) and \(\Phi :(m^{\ast }, n^{\ast })\rightarrow \mathbb{R}\) is convex, then

$$ \Phi \biggl( \frac{\int _{u}^{v}\int _{w}^{s}f(r^{\ast }, t^{\ast })\Delta _{1}r^{\ast }\Delta _{2}t^{\ast }}{\int _{u}^{v}\int _{w}^{s}\Delta _{1}r^{\ast }\Delta _{2}t^{\ast }} \biggr) \leq \frac{\int _{u}^{v}\int _{w}^{s}\Phi (f(r^{\ast }, t^{\ast }))\Delta _{1}r^{\ast }\Delta _{2}t^{\ast }}{\int _{u}^{v}\int _{w}^{s}\Delta _{1}r^{\ast }\Delta _{2}t^{\ast }}. $$
(9)

Lemma 1

(Young’s inequality [25])

If \(l, u\in \mathbb{R}_{+}\) and \(p^{-1}+q^{-1}=1\) with \(p>1\), then

$$ lu\leq p^{-1}l^{p}+q^{-1}u^{q}, $$
(10)

we get equality iff \(l^{p}=u^{q}\).

Main results

In this section, we state and prove our main results. In particular, we establish the two-dimensional versions of the inequalities given in [7]. Throughout this section, we will assume that the following hypotheses hold:

(H1) \(\mathbb{T}_{1}\) and \(\mathbb{T}_{2}\) are any two time scales with (i) \(t_{0}, s_{1}, k_{1}, x, z\in \mathbb{T}_{1}\); (ii) \(t_{0}, t_{1}, r_{1}, y, w\in \mathbb{T}_{2}\).

(H2) \(p_{1}\), \(q_{1}\) are any two real numbers such that \(p_{1}>1\), \(q_{1}>1\) with \(1/p_{1}+1/q_{1}=1\).

(H3) For \(t_{0}\in \mathbb{T}_{1}, \mathbb{T}_{2}\) we denote the subintervals of \(\mathbb{T}_{1}\), \(\mathbb{T}_{2}\) by \(I_{x}=[t_{0}, x)_{\mathbb{T}_{1}}\), \(I_{z}=[t_{0}, z)_{\mathbb{T}_{1}}\), \(I_{y}=[t_{0}, y)_{\mathbb{T}_{2}}\) and \(I_{w}=[t_{0}, w)_{\mathbb{T}_{2}}\), where \(x, z\in \Omega _{1}=[t_{0}, \infty )\cap \mathbb{T}_{1}\) and \(y,w\in \Omega _{2}=[t_{0}, \infty )\cap \mathbb{T}_{2}\).

(H4) There exist two functions Φ and Ψ which are real-valued, nonnegative, convex, and submultiplicative, defined on \([ 0, \infty )\). A function \(f^{\ast }\) is submultiplicative if \(f^{\ast }(x_{1}y_{1})\leq f^{\ast }(x_{1})f^{\ast }(y_{1})\) for \(x_{1}, y_{1}\geq 0\).

Theorem 4

Let (H1), (H2) be satisfied and \(f^{\ast }(s_{1}, t_{1})\in \mathrm{CC}_{rd}^{1}(I_{x}\times I_{y},\mathbb{R}^{+})\), \(g^{\ast }(k_{1}, r_{1})\in \mathrm{CC}_{rd}^{1}(I_{z}\times I_{w}, \mathbb{R}^{+})\). Suppose that \(F^{\ast }(s_{1}, t_{1})\) and \(G^{\ast }(k_{1}, r_{1})\) are defined as

$$ F^{\ast }(s_{1}, t_{1})= \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}}f^{\ast }(\xi , \eta )\Delta \xi \Delta \eta , G^{\ast }(k_{1},r_{1})= \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}g^{\ast }( \xi , \eta )\Delta \xi \Delta \eta . $$
(11)

Then for \((s_{1}, t_{1})\in I_{x}\times I_{y}\) and \((k_{1}, r_{1})\in I_{z}\times I_{w}\), one gets

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{F^{\ast }(s_{1}, t_{1})G^{\ast }(k_{1}, r_{1})}{q_{1} [ (s_{1}-t_{0})(t_{1}-t_{0}) ] ^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{{q_{1}-1}}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq C(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl( \sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[ f^{\ast }(s_{1}, t_{1}) \bigr] ^{p_{1}}\Delta s_{1}\Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl(\sigma (w)-k_{1} \bigr) \bigl( \sigma (z)-r_{1}\bigr) \bigl[ g^{\ast }(k_{1}, r_{1}) \bigr] ^{q_{1}} \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}}, \end{aligned}$$
(12)

where

$$ C(p_{1}, q_{1})=\frac{1}{p_{1}q_{1}} \bigl[ (x-t_{0}) (y-t_{0}) \bigr] ^{\frac{p_{1}-1}{p_{1}}} \bigl[ (w-t_{0}) (z-t_{0}) \bigr] ^{ \frac{q_{1}-1}{q_{1}}}. $$
(13)

Proof

By assumption and applying Hölder’s inequality (8) with respect to \(p_{1}\), \(p_{1}/(p_{1}-1)\) and \(q_{1}\), \(q_{1}/(q_{1}-1)\), respectively, we find that

$$ F^{\ast }(s_{1}, t_{1})\leq \bigl[ (s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] ^{\frac{p_{1}-1}{p_{1}}} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ f^{\ast }(\xi , \eta ) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} $$
(14)

and

$$ G^{\ast }(k_{1}, r_{1})\leq \bigl[ (k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] ^{\frac{q_{1}-1}{q_{1}}} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ g^{\ast }(\xi , \eta ) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}}. $$
(15)

By multiplying (14) and (15), we get

$$\begin{aligned} F^{\ast }(s_{1}, t_{1})G^{\ast }(k_{1}, r_{1}) \leq & \bigl[ (s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] ^{\frac{p_{1}-1}{p_{1}}} \bigl[ (k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] ^{\frac{q_{1}-1}{q_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ f^{ \ast }(\xi , \eta ) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ g^{ \ast }(\xi , \eta ) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$
(16)

Applying Young’s inequality on the term \([ (s_{1}-t_{0})(t_{1}-t_{0}) ] ^{(p_{1}-1)/p_{1}}\times [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)/q_{1}}\) with \(u= [ (s_{1}-t_{0})(t_{1}-t_{0}) ] ^{(p_{1}-1)/p_{1}}\) and \(l= [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)/q_{1}}\), we observe that

$$\begin{aligned} F^{\ast }(s_{1}, t_{1})G^{\ast }(k_{1}, r_{1}) \leq & \biggl( \frac{[(s_{1}-t_{0})(t_{1}-t_{0})]^{{(p_{1}-1)}}}{p_{1}}+ \frac{ [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)}}{q_{1}} \biggr) \\ &{}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ f^{ \ast }(\xi , \eta ) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ g^{ \ast }(\xi , \eta ) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}} \\ =& \biggl( \frac{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{(p_{1}-1)}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)}}{p_{1}q_{1}} \biggr) \\ &{}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ f^{ \ast }(\xi , \eta ) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ g^{ \ast }(\xi , \eta ) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$
(17)

Dividing both sides of (17) by \(q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{(p_{1}-1)}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)}\), we obtain

$$\begin{aligned} &\frac{F^{\ast }(s_{1}, t_{1})G^{\ast }(k_{1}, r_{1})}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{(p_{1}-1)}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)}} \\ &\quad \leq \frac{1}{p_{1}q_{1}} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ f^{\ast }(\xi , \eta ) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ g^{ \ast }(\xi , \eta ) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$
(18)

Integrating both sides of (18) first with respect to \(r_{1}\) and \(k_{1} \) and then with respect to \(s_{1}\) and \(t_{1}\), respectively, and applying Hölder’s inequality (8) with indices \(p_{1}\), \(p_{1}/(p_{1}-1)\) and \(q_{1}\), \(q_{1}/(q_{1}-1)\), we see that

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{F^{\ast }(s_{1}, t_{1})G^{\ast }(k_{1}, r_{1})}{q_{1} [ (s_{1}-t_{0})(t_{1}-t_{0}) ] ^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{{q_{1}-1}}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq \frac{1}{p_{1}q_{1}} \bigl[ (x-t_{0}) (y-t_{0}) \bigr] ^{ \frac{p_{1}-1}{p_{1}}} \bigl[ (z-t_{0}) (w-t_{0}) \bigr] ^{\frac{q_{1}-1}{q_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ f^{\ast }(\xi , \eta ) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) \Delta s_{1}\Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ g^{\ast }(\xi , \eta ) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}} \\ &\quad =C(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ f^{\ast }( \xi , \eta ) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) \Delta s_{1} \Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ g^{\ast }(\xi , \eta ) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$
(19)

Applying Fubini’s theorem on (19), we conclude that

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{F^{\ast }(s_{1}, t_{1})G^{\ast }(k_{1}, r_{1})}{q_{1} [ (s_{1}-t_{0})(t_{1}-t_{0}) ] ^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{{q_{1}-1}}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq C(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}(x-s_{1}) (y-t_{1}) \bigl[ f^{\ast }(s_{1},t_{1}) \bigr] ^{p_{1}}\Delta s_{1}\Delta t_{1} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}(z-k_{1}) (w-r_{1}) \bigl[ g^{\ast }(k_{1}, r_{1}) \bigr] ^{q_{1}}\Delta k_{1} \Delta r_{1} \biggr) ^{\frac{1}{q_{1}}}, \end{aligned}$$

and then, by using the fact that \(\sigma (n)\geq n\), one gets

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{F^{\ast }(s_{1}, t_{1})G^{\ast }(k_{1}, r_{1})}{q_{1} [ (s_{1}-t_{0})(t_{1}-t_{0}) ] ^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{{q_{1}-1}}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq C(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl( \sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[ f^{\ast }(s_{1}, t_{1}) \bigr] ^{p_{1}}\Delta s_{1}\Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl(\sigma (z)-r_{1} \bigr) \bigl( \sigma (w)-k_{1}\bigr) \bigl[ g^{\ast }(k_{1}, r_{1}) \bigr] ^{q_{1}} \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}}, \end{aligned}$$

which proves (12). This completes the proof. □

Using the relations (5) and taking \(\mathbb{T}_{1}= \mathbb{T}_{2}=\mathbb{R}\), \(t_{0}=0\) in Theorem 4 leads to the following result.

Corollary 1

Assume that \(f^{\ast }(s_{1}, t_{1})\) and \(g^{\ast }(k_{1}, r_{1})\) are real-valued continuous functions and define

$$ F^{\ast }(s_{1}, t_{1})= \int _{0}^{s_{1}} \int _{0}^{t_{1}}f^{ \ast }(\xi , \eta )\,d\xi \,d\eta ,\qquad G^{\ast }(k_{1}, r_{1})= \int _{0}^{k_{1}} \int _{0}^{r_{1}}g^{\ast }(\xi , \eta )\,d \xi \,d\eta . $$

Then for \((s_{1}, t_{1})\in I_{x}\times I_{y}\) and \((k_{1}, r_{1})\in I_{z}\times I_{w}\), we have

$$\begin{aligned} & \int _{0}^{x} \int _{0}^{y} \biggl( \int _{0}^{z} \int _{0}^{w} \frac{F^{\ast }(s_{1}, t_{1})G^{\ast }(k_{1}, r_{1})}{q_{1}(s_{1}t_{1})^{{p_{1}-1}}+p_{1}(k_{1}r_{1})^{{q_{1}-1}}}\,dk_{1} \,dr_{1} \biggr) \,ds_{1}\,dt_{1} \\ &\quad \leq C^{\ast }(p_{1}, q_{1}) \biggl( \int _{0}^{x} \int _{0}^{y}(x-s_{1}) (y-t_{1}) \bigl[ f^{\ast }(s_{1}, t_{1}) \bigr] ^{p_{1}}\,ds_{1}\,dt_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{0}^{z} \int _{0}^{w}(z-k_{1}) (w-r_{1}) \bigl[ g^{ \ast }(k_{1}, r_{1}) \bigr] ^{q_{1}}\,dk_{1}\,dr_{1} \biggr) ^{ \frac{1}{q_{1}}}, \end{aligned}$$

where

$$ C^{\ast }(p_{1}, q_{1})=\frac{1}{p_{1}q_{1}}(xy)^{ \frac{p_{1}-1}{p_{1}}}(zw)^{\frac{q_{1}-1}{q_{1}}}. $$

By using the relations (5) and letting \(\mathbb{T}_{1}= \mathbb{T}_{2}=\mathbb{Z}\), \(t_{0}=0\) in Theorem 4, we get the following result.

Corollary 2

Assume that \(\{a_{m_{1}, n_{1}}\}_{0\leq m_{1}, n_{1}\leq N}\) and \(\{b_{k_{1}, r_{1}}\}_{0\leq k_{1}, r_{1}\leq N}\) are two nonnegative sequences of real numbers and define

$$ A_{{m_{1}, n_{1}}}=\sum_{\xi =1}^{m_{1}}\sum _{\eta =1}^{n_{1}}a_{ \xi , \eta },\qquad B_{{k_{1}, r_{1}}}=\sum_{\xi =1}^{k_{1}} \sum _{\eta =1}^{r_{1}}b_{\xi , \eta }. $$

Then

$$\begin{aligned} &\sum_{s_{1}=1}^{m_{1}}\sum _{t_{1}=1}^{n_{1}} \Biggl( \sum _{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{w_{1}} \frac{A_{{s_{1}, t_{1}}}B_{{k_{1}, r_{1}}}}{q_{1}(s_{1}t_{1})^{{p_{1}-1}}+p_{1}(k_{1}r_{1})^{{q_{1}-1}}} \Biggr) \\ &\quad \leq C^{\ast \ast }(p_{1}, q_{1}) \Biggl( \sum _{s_{1}=1}^{m_{1}}\sum _{t_{1}=1}^{n_{1}}(m_{1}-s_{1}+1) (n_{1}-t_{1}+1) (a_{s_{1}, t_{1}})^{p_{1}} \Biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \Biggl( \sum_{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{w_{1}}(z_{1}-k_{1}+1) (w_{1}-r_{1}+1) (b_{k_{1}, r_{1}})^{q_{1}} \Biggr) ^{\frac{1}{q_{1}}}, \end{aligned}$$

where

$$ C^{\ast \ast }(p_{1}, q_{1})=\frac{1}{p_{1}q_{1}}(m_{1}n_{1})^{ \frac{p_{1}-1}{p_{1}}}(z_{1}w_{1})^{\frac{q_{1}-1}{q_{1}}}. $$

In the following theorems, we give a further generalization of (12) obtained in Theorem 4.

Theorem 5

Let (H1), (H2), and (H4) be satisfied, \(f^{\ast }(s_{1}, t_{1})\in \mathrm{CC}_{rd}^{1}(I_{x}\times I_{y}, \mathbb{R}^{+})\), \(g^{\ast }(k_{1}, r_{1})\in \mathrm{CC}_{rd}^{1}(I_{z}\times I_{w}, \mathbb{R}^{+})\) and \(p^{\ast }(\xi , \eta )\), \(q^{\ast }(\xi , \eta )\) be two positive functions. Suppose that \(F^{\ast }(s_{1}, t_{1})\) and \(G^{\ast }(k_{1}, r_{1})\) are as defined in Theorem 4and let

$$ P^{\ast }(s_{1}, t_{1})= \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}}p^{\ast }(\xi , \eta )\Delta \xi \Delta \eta ,\qquad Q^{\ast }(k_{1},r_{1})= \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}q^{\ast }( \xi , \eta )\Delta \xi \Delta \eta . $$
(20)

Then for \((s_{1}, t_{1})\in I_{x}\times I_{y}\) and \((k_{1}, r_{1})\in I_{z}\times I_{w}\), we have

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))}{q_{1} [ (s_{1}-t_{0})(t_{1}-t_{0}) ] ^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{{q_{1}-1}}}\Delta k_{1} \Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq D(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl( \sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \biggl( p^{\ast }(s_{1}, t_{1}) \Phi \biggl[ \frac{f^{\ast }(s_{1}, t_{1})}{p^{\ast }(s_{1}, t_{1})} \biggr] \biggr) ^{p_{1}} \Delta s_{1}\Delta t_{1} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (w)-r_{1}\bigr) \biggl( q^{\ast }(k_{1}, r_{1})\Psi \biggl[ \frac{g^{\ast }(k_{1}, r_{1})}{q^{\ast }(k_{1}, r_{1})} \biggr] \biggr) ^{q_{1}} \Delta k_{1}\Delta r_{1} \biggr) ^{ \frac{1}{q_{1}}}, \end{aligned}$$
(21)

where

$$\begin{aligned} D(p_{1}, q_{1}) =&\frac{1}{p_{1}q_{1}} \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \frac{\Phi (P^{\ast }(s_{1}, t_{1}))}{P^{\ast }(s_{1}, t_{1})} \biggr) ^{ \frac{p_{1}}{p_{1}-1}}\Delta s_{1}\Delta t_{1} \biggr) ^{\frac{p_{1}-1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \biggl( \frac{\Psi (Q^{\ast }(k_{1}, r_{1}))}{Q^{\ast }(k_{1}, r_{1})} \biggr) ^{\frac{q_{1}}{q_{1}-1}}\Delta k_{1}\Delta r_{1} \biggr) ^{\frac{q_{1}-1}{q_{1}}}. \end{aligned}$$
(22)

Proof

By assumption and using the Jensen’s inequality (9), it is clear that

$$\begin{aligned} \Phi \bigl(F^{\ast }(s_{1},t_{1})\bigr) =&\Phi \biggl( \frac{P^{\ast }(s_{1}, t_{1})\int _{t_{0}}^{s_{1}}\int _{t_{0}}^{t_{1}}p^{\ast }(\xi , \eta ) [ \frac{f^{\ast }(\xi , \eta )}{p^{\ast }(\xi , \eta )} ] \Delta \xi \Delta \eta }{\int _{t_{0}}^{s_{1}}\int _{t_{0}}^{t_{1}}p^{\ast }(\xi , \eta )\Delta \xi \Delta \eta } \biggr) \\ \leq &\Phi \bigl(P^{\ast }(s_{1}, t_{1})\bigr)\Phi \biggl( \frac{\int _{t_{0}}^{s_{1}}\int _{t_{0}}^{t_{1}}p^{\ast }(\xi , \eta ) [ \frac{f^{\ast }(\xi , \eta )}{p^{\ast }(\xi , \eta )} ] \Delta \xi \Delta \eta }{\int _{t_{0}}^{s_{1}}\int _{t_{0}}^{t_{1}}p^{\ast }(\xi , \eta )\Delta \xi \Delta \eta } \biggr) \\ \leq &\frac{\Phi (P^{\ast }(s_{1}, t_{1}))}{P^{\ast }(s_{1}, t_{1})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}}p^{\ast }(\xi , \eta )\Phi \biggl[ \frac{f^{\ast }(\xi , \eta )}{p^{\ast }(\xi , \eta )} \biggr] \Delta \xi \Delta \eta . \end{aligned}$$
(23)

Applying Hölder’s inequality (8) with indices \(p_{1}\) and \(p_{1}/(p_{1}-1)\) on the right-hand side of (23), we have

$$\begin{aligned} \Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr) \leq & \bigl[ (s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] ^{\frac{p_{1}-1}{p_{1}}} \frac{\Phi (P^{\ast }(s_{1}, t_{1}))}{P^{\ast }(s_{1}, t_{1})} \\ &{}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl( p^{ \ast }(\xi , \eta ) \Phi \biggl[ \frac{f^{\ast }(\xi , \eta )}{p^{\ast }(\xi , \eta )} \biggr] \biggr) ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{ \frac{1}{p_{1}}}. \end{aligned}$$
(24)

Analogously,

$$\begin{aligned} \Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr) \leq & \bigl[ (k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] ^{\frac{q_{1}-1}{q_{1}}} \frac{\Psi (Q^{\ast }(k_{1}, r_{1}))}{Q^{\ast }(k_{1}, r_{1})} \\ &{}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl( q^{ \ast }(\xi , \eta ) \Psi \biggl[ \frac{g^{\ast }(\xi , \eta )}{q^{\ast }(\xi , \eta )} \biggr] \biggr) ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{ \frac{1}{q_{1}}}. \end{aligned}$$
(25)

Thus, from (24) and (25), it can be concluded that

$$\begin{aligned} &\Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr)\Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr) \\ &\quad \leq \bigl[ (s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] ^{ \frac{p_{1}-1}{p_{1}}} \bigl[ (k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] ^{\frac{q_{1}-1}{q_{1}}} \\ &\qquad {}\times \biggl( \frac{\Phi (P^{\ast }(s_{1}, t_{1}))}{P^{\ast }(s_{1}, t_{1})} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl( p^{\ast }( \xi , \eta )\Phi \biggl[ \frac{f^{\ast }(\xi , \eta )}{p^{\ast }(\xi , \eta )} \biggr] \biggr) ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \biggr) \\ &\qquad {}\times \biggl( \frac{\Psi (Q^{\ast }(k_{1}, r_{1}))}{Q^{\ast }(k_{1}, r_{1})} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl( q^{\ast }( \xi , \eta )\Psi \biggl[ \frac{g^{\ast }(\xi , \eta )}{q^{\ast }(\xi , \eta )} \biggr] \biggr) ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}} \biggr) . \end{aligned}$$
(26)

Applying Young’s inequality on the term \([ (s_{1}-t_{0})(t_{1}-t_{0}) ] ^{(p_{1}-1)/p_{1}} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)/q_{1}}\), we get

$$\begin{aligned} &\Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr)\Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr) \\ &\quad \leq \biggl( \frac{[(s_{1}-t_{0})(t_{1}-t_{0})]^{{(p_{1}-1)}}}{p_{1}}+\frac{ [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)}}{q_{1}} \biggr) \\ &\qquad {}\times \biggl( \frac{\Phi (P^{\ast }(s_{1}, t_{1}))}{P^{\ast }(s_{1}, t_{1})} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl( p^{\ast }( \xi , \eta )\Phi \biggl[ \frac{f^{\ast }(\xi , \eta )}{p^{\ast }(\xi , \eta )} \biggr] \biggr) ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \biggr) \\ &\qquad {}\times \biggl( \frac{\Psi (Q^{\ast }(k_{1}, r_{1}))}{Q^{\ast }(k_{1}, r_{1})} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl( q^{\ast }( \xi , \eta )\Psi \biggl[ \frac{g^{\ast }(\xi , \eta )}{q^{\ast }(\xi , \eta )} \biggr] \biggr) ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}} \biggr) . \end{aligned}$$
(27)

From (27), we observe that

$$\begin{aligned} &\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{(p_{1}-1)}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)}} \\ &\quad \leq \frac{1}{p_{1}q_{1}} \biggl( \frac{\Phi (P^{\ast }(s_{1}, t_{1}))}{P^{\ast }(s_{1}, t_{1})} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl( p^{\ast }(\xi , \eta ) \Phi \biggl[ \frac{f^{\ast }(\xi , \eta )}{p^{\ast }(\xi , \eta )} \biggr] \biggr) ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{ \frac{1}{p_{1}}} \biggr) \\ &\qquad {}\times \biggl( \frac{\Psi (Q^{\ast }(k_{1}, r_{1}))}{Q^{\ast }(k_{1}, r_{1})} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl( q^{\ast }( \xi , \eta )\Psi \biggl[ \frac{g^{\ast }(\xi , \eta )}{q^{\ast }(\xi , \eta )} \biggr] \biggr) ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}} \biggr) . \end{aligned}$$
(28)

Integrating both sides of (28) first with respect to \(r_{1}\) and \(k_{1}\) and then with respect to \(s_{1}\) and \(t_{1}\), respectively, we get

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{(p_{1}-1)}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq \frac{1}{p_{1}q_{1}} \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \frac{\Phi (P^{\ast }(s_{1}, t_{1}))}{P^{\ast }(s_{1}, t_{1})} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl( p^{\ast }( \xi , \eta )\Phi \biggl[ \frac{f^{\ast }(\xi , \eta )}{p^{\ast }(\xi , \eta )} \biggr] \biggr) ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{ \frac{1}{p_{1}}}\Delta s_{1}\Delta t_{1} \biggr) \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \frac{\Psi (Q^{\ast }(k_{1},r_{1}))}{Q^{\ast }(k_{1}, r_{1})} \\ &\qquad {}\times\biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl( q^{\ast }(\xi , \eta ) \Psi \biggl[ \frac{g^{\ast }(\xi , \eta )}{q^{\ast }(\xi , \eta )} \biggr] \biggr) ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{ \frac{1}{q_{1}}}\Delta k_{1}\Delta r_{1} \biggr) . \end{aligned}$$
(29)

Using Hölder’s inequality (8) again with respect to \(p_{1}\), \(p_{1}/(p_{1}-1)\) and \(q_{1}\), \(q_{1}/(q_{1}-1)\), respectively, on (29), we may write

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{(p_{1}-1)}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq \frac{1}{p_{1}q_{1}} \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \frac{\Phi (P^{\ast }(s_{1}, t_{1}))}{P^{\ast }(s_{1}, t_{1})} \biggr) ^{\frac{p_{1}}{p_{1}-1}}\Delta s_{1}\Delta t_{1} \biggr) ^{ \frac{p_{1}-1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \biggl( \frac{\Psi (Q^{\ast }(k_{1}, r_{1}))}{Q^{\ast }(k_{1}, r_{1})} \biggr) ^{\frac{q_{1}}{q_{1}-1}}\Delta k_{1}\Delta r_{1} \biggr) ^{\frac{q_{1}-1}{q_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl( p^{\ast }(\xi , \eta ) \Phi \biggl[ \frac{f^{\ast }(\xi , \eta )}{p^{\ast }(\xi , \eta )} \biggr] \biggr) ^{p_{1}}\Delta \xi \Delta \eta \biggr) \Delta s_{1}\Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl( q^{\ast }(\xi , \eta ) \Psi \biggl[ \frac{g^{\ast }(\xi , \eta )}{q^{\ast }(\xi , \eta )} \biggr] \biggr) ^{q_{1}}\Delta \xi \Delta \eta \biggr) \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}} \\ &\quad =D(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl( p^{\ast }( \xi , \eta )\Phi \biggl[ \frac{f^{\ast }(\xi , \eta )}{p^{\ast }(\xi , \eta )} \biggr] \biggr) ^{p_{1}}\Delta \xi \Delta \eta \biggr) \Delta s_{1}\Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl( q^{\ast }(\xi , \eta ) \Psi \biggl[ \frac{g^{\ast }(\xi , \eta )}{q^{\ast }(\xi , \eta )} \biggr] \biggr) ^{q_{1}}\Delta \xi \Delta \eta \biggr) \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$

Applying Fubini’s theorem and using the fact that \(\sigma (n)\geq n\), we get

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{(p_{1}-1)}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq D(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}(x-s_{1}) (y-t_{1}) \biggl( p^{\ast }(s_{1},t_{1}) \Phi \biggl[ \frac{f^{\ast }(s_{1}, t_{1})}{p^{\ast }(s_{1}, t_{1})} \biggr] \biggr) ^{p_{1}}\Delta s_{1}\Delta t_{1} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}(z-k_{1}) (w-r_{1}) \biggl( q^{\ast }(k_{1}, r_{1}) \Psi \biggl[ \frac{g^{\ast }(k_{1}, r_{1})}{q^{\ast }(k_{1}, r_{1})} \biggr] \biggr) ^{q_{1}} \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}} \\ &\quad \leq D(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl( \sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \biggl( p^{\ast }(s_{1}, t_{1}) \Phi \biggl[ \frac{f^{\ast }(s_{1}, t_{1})}{p^{\ast }(s_{1}, t_{1})} \biggr] \biggr) ^{p_{1}} \Delta s_{1}\Delta t_{1} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (w)-r_{1}\bigr) \biggl( q^{\ast }(k_{1}, r_{1})\Psi \biggl[ \frac{g^{\ast }(k_{1}, r_{1})}{q^{\ast }(k_{1}, r_{1})} \biggr] \biggr) ^{q_{1}} \Delta k_{1}\Delta r_{1} \biggr) ^{ \frac{1}{q_{1}}}, \end{aligned}$$

which is (21). This completes the proof. □

By using the relations (5) and taking \(\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{R}\), \(t_{0}=0\) in Theorem 5, we get the following result.

Corollary 3

Assume that \(f^{\ast }(s_{1}, t_{1})\), \(g^{\ast }(k_{1}, r_{1})\) are real-valued continuous functions, \(p^{\ast }(s_{1}, t_{1})\), \(q^{\ast }(k_{1}, r_{1})\) are two positive functions, and define

$$\begin{aligned} &F^{\ast }(s_{1}, t_{1}) = \int _{0}^{s_{1}} \int _{0}^{t_{1}}f^{ \ast }(\xi , \eta )\,d\xi \,d\eta ,\qquad G^{\ast }(k_{1}, r_{1})= \int _{0}^{k_{1}} \int _{0}^{r_{1}}g^{\ast }(\xi , \eta )\,d \xi \,d\eta , \\ &P^{\ast }(s_{1}, t_{1}) = \int _{0}^{s_{1}} \int _{0}^{t_{1}}p^{ \ast }(\xi , \eta )\,d\xi \,d\eta ,\qquad Q^{\ast }(k_{1}, r_{1})= \int _{0}^{k_{1}} \int _{0}^{r_{1}}q^{\ast }(\xi , \eta )\,d \xi \,d\eta . \end{aligned}$$

Then for \((s_{1}, t_{1})\in I_{x}\times I_{y}\) and \((k_{1}, r_{1})\in I_{z}\times I_{w}\), we have

$$\begin{aligned} & \int _{0}^{x} \int _{0}^{y} \biggl( \int _{0}^{z} \int _{0}^{w} \frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))}{q_{1}(s_{1}t_{1})^{{p_{1}-1}}+p_{1}(k_{1}r_{1})^{{q_{1}-1}}}\,dk_{1} \,dr_{1} \biggr) \,ds_{1}\,dt_{1} \\ &\quad \leq D^{\ast }(p_{1}, q_{1}) \biggl( \int _{0}^{x} \int _{0}^{y}(x-s_{1}) (y-t_{1}) \biggl( p^{\ast }(s_{1}, t_{1}) \Phi \biggl[ \frac{f^{\ast }(s_{1}, t_{1})}{p^{\ast }(s_{1},t_{1})} \biggr] \biggr) ^{p_{1}}\,ds_{1} \,dt_{1} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{0}^{z} \int _{0}^{w}(z-k_{1}) (w-r_{1}) \biggl( q^{ \ast }(k_{1}, r_{1}) \Psi \biggl[ \frac{g^{\ast }(k_{1}, r_{1})}{q^{\ast }(k_{1}, r_{1})} \biggr] \biggr) ^{q_{1}}\,dk_{1} \,dr_{1} \biggr) ^{\frac{1}{q_{1}}}, \end{aligned}$$

where

$$\begin{aligned} D^{\ast }(p_{1}, q_{1}) &=\frac{1}{p_{1}q_{1}} \biggl( \int _{0}^{x} \int _{0}^{y} \biggl( \frac{\Phi (P^{\ast }(s_{1}, t_{1}))}{P^{\ast }(s_{1}, t_{1})} \biggr) ^{\frac{p_{1}}{p_{1}-1}}\,ds_{1}\,dt_{1} \biggr) ^{\frac{p_{1}-1}{p_{1}}} \\ &\quad {}\times \biggl( \int _{0}^{z} \int _{0}^{w} \biggl( \frac{\Psi (Q^{\ast }(k_{1},r_{1}))}{Q^{\ast }(k_{1}, r_{1})} \biggr) ^{ \frac{q_{1}}{q_{1}-1}}\,dk_{1}\,dr_{1} \biggr) ^{\frac{q_{1}-1}{q_{1}}}. \end{aligned}$$

By using the relations (5) and taking \(\mathbb{T}_{1}= \mathbb{T}_{2}=\mathbb{Z}\), \(t_{0}=0\) in Theorem 5, we get the following result.

Corollary 4

Assume that \(\{a_{m_{1}, n_{1}}\}_{0\leq m_{1}, n_{1}\leq N}\), \(\{b_{k_{1}, r_{1}}\}_{0\leq k_{1}, r_{1}\leq N}\) are two nonnegative sequences of real numbers, \(\{p_{m_{1}, n_{1}}\}_{{0\leq m_{1}, n_{1}\leq N}}\), \(\{q_{k_{1}, r_{1}}\}_{0\leq k_{1}, r_{1}\leq N}\) are positive sequences, and define

$$\begin{aligned} &A_{{m_{1}, n_{1}}} =\sum_{\xi =1}^{m_{1}}\sum _{\eta =1}^{n_{1}}a_{ \xi , \eta },\qquad B_{{k_{1}, r_{1}}}=\sum_{\xi =1}^{k_{1}}\sum _{\eta =1}^{r_{1}}b_{\xi , \eta }, \\ &P_{m_{1}, n_{1}} =\sum_{\xi =1}^{m_{1}}\sum _{\eta =1}^{n_{1}}p_{ \xi , \eta },\qquad Q_{{k_{1}, r_{1}}}=\sum_{\xi =1}^{k_{1}} \sum _{\eta =1}^{r_{1}}q_{\xi , \eta }. \end{aligned}$$

Then

$$\begin{aligned} &\sum_{s_{1}=1}^{m_{1}}\sum _{t_{1}=1}^{n_{1}} \Biggl( \sum _{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{w_{1}} \frac{\Phi (A_{{s_{1}, t_{1}}})\Psi (B_{k_{1}, r_{1}})}{q_{1}(s_{1}t_{1})^{{p_{1}-1}}+p_{1}(k_{1}r_{1})^{{q_{1}-1}}} \Biggr) \\ &\quad \leq D^{\ast \ast }(p_{1}, q_{1}) \Biggl( \sum _{s_{1}=1}^{m_{1}} \sum _{t_{1}=1}^{n_{1}}(m_{1}-s_{1}+1) (n_{1}-t_{1}+1) \biggl( p_{s_{1}, t_{1}}\Phi \biggl[ \frac{a_{s_{1}, t_{1}}}{p_{s_{1}, t_{1}}} \biggr] \biggr) ^{p_{1}} \Biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad {}\times \Biggl( \sum_{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{w_{1}}(z_{1}-k_{1}+1) (w_{1}-r_{1}+1) \biggl( q_{k_{1}, r_{1}}\Psi \biggl[ \frac{b_{{k_{1}, r_{1}}}}{q_{k_{1}, r_{1}}} \biggr] \biggr) ^{q_{1}} \Biggr) ^{ \frac{1}{q_{1}}}, \end{aligned}$$

where

$$ D^{\ast \ast }(p_{1}, q_{1})=\frac{1}{p_{1}q_{1}} \Biggl( \sum_{s_{1}=1}^{m_{1}} \sum _{t_{1}=1}^{n_{1}} \biggl( \frac{\Phi (P_{s_{1},t_{1}})}{P_{s_{1}, t_{1}}} \biggr) ^{ \frac{p_{1}}{p_{1}-1}} \Biggr) ^{\frac{p_{1}-1}{p_{1}}} \Biggl( \sum _{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{w_{1}} \biggl( \frac{\Psi (Q_{k_{1},r_{1}})}{Q_{k_{1}, r_{1}}} \biggr) ^{ \frac{q_{1}}{q_{1}-1}} \Biggr) ^{\frac{q_{1}-1}{q_{1}}}. $$

Remark 1

By applying (10) on (12) in Theorem 4 and (21) in Theorem 5, respectively, we get the following inequalities:

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{F^{\ast }(s_{1}, t_{1})G^{\ast }(k_{1}, r_{1})}{q_{1} [ (s_{1}-t_{0})(t_{1}-t_{0}) ] ^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{{q_{1}-1}}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq C(p_{1}, q_{1}) \biggl\{ \frac{1}{p_{1}} \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[ f^{\ast }(s_{1}, t_{1}) \bigr] ^{p_{1}}\Delta s_{1}\Delta t_{1} \biggr) \\ &\qquad {} +\frac{1}{q_{1}} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl( \sigma (z)-k_{1} \bigr) \bigl(\sigma (w)-r_{1}\bigr) \bigl[ g^{\ast }(k_{1}, r_{1}) \bigr] ^{q_{1}}\Delta k_{1}\Delta r_{1} \biggr) \biggr\} , \end{aligned}$$
(30)

where \(C(p_{1}, q_{1})\) is defined in (13), and

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{(p_{1}-1)}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq D(p_{1}, q_{1}) \biggl\{ \frac{1}{p_{1}} \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \biggl( p^{\ast }(s_{1}, t_{1})\Phi \biggl[ \frac{f^{\ast }(s_{1}, t_{1})}{p^{\ast }(s_{1}, t_{1})} \biggr] \biggr) ^{p_{1}} \Delta s_{1}\Delta t_{1} \biggr) \\ &\qquad {} +\frac{1}{q_{1}} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl( \sigma (z)-k_{1} \bigr) \bigl(\sigma (w)-r_{1}\bigr) \\ &\qquad {}\times \biggl( q^{\ast }(k_{1}, r_{1}) \Psi \biggl[ \frac{g^{\ast }(k_{1}, r_{1})}{q^{\ast }(k_{1}, r_{1})} \biggr] \biggr) ^{q_{1}} \Delta k_{1}\Delta r_{1} \biggr) \biggr\} , \end{aligned}$$
(31)

where \(D(p_{1}, q_{1})\) is defined in (22).

The following theorems present slight variants of (21) in Theorem 5.

Theorem 6

Let (H1), (H2), and (H4) be satisfied and \(f^{\ast }(s_{1}, t_{1})\in \mathrm{CC}_{rd}^{1}(I_{x}\times I_{y}, \mathbb{R}^{+})\), \(g^{\ast }(k_{1}, r_{1})\in \mathrm{CC}_{rd}^{1}(I_{z}\times I_{w}, \mathbb{R}^{+})\). Define

$$\begin{aligned} &F^{\ast }(s_{1}, t_{1}) =\frac{1}{(s_{1}-t_{0})(t_{1}-t_{0})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}}f^{\ast }(\xi , \eta ) \Delta \xi \Delta \eta , \\ &G^{\ast }(k_{1}, r_{1}) =\frac{1}{(k_{1}-t_{0})(r_{1}-t_{0})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}g^{\ast }(\xi , \eta ) \Delta \xi \Delta \eta . \end{aligned}$$
(32)

Then for \((s_{1}, t_{1})\in I_{x}\times I_{y}\) and \((k_{1}, r_{1})\in I_{z}\times I_{w}\), one gets

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0})}{q_{1} [ (s-t_{0})(t-t_{0}) ] ^{{p_{1}-1}}+p_{1} [ (k-t_{0})(r-t_{0}) ] ^{{q_{1}-1}}} \\ &\qquad {}\times\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq K(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl( \sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[ \Phi \bigl(f^{\ast }(s_{1}, t_{1})\bigr) \bigr] ^{p_{1}}\Delta s_{1}\Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl(\sigma (w)-k_{1} \bigr) \bigl( \sigma (z)-r_{1}\bigr) \bigl[ \Psi \bigl(g^{\ast }(k_{1}, r_{1})\bigr) \bigr] ^{q_{1}} \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}}, \end{aligned}$$
(33)

where

$$ K(p_{1}, q_{1})=\frac{1}{p_{1}q_{1}}\bigl[(s_{1}-t_{0}) (t_{1}-t_{0})\bigr]^{\frac{p_{1}-1}{p_{1}}}\bigl[(k_{1}-t_{0}) (r_{1}-t_{0})\bigr]^{ \frac{q_{1}-1}{q_{1}}}. $$
(34)

Proof

By assumption and using the Jensen’s inequality (9), we obtain

$$\begin{aligned} \Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr) =&\Phi \biggl( \frac{1}{(s_{1}-t_{0})(t_{1}-t_{0})} \int _{t}^{s_{1}} \int _{t_{0}}^{t_{1}}f^{ \ast }(\xi , \eta )\Delta \xi \Delta \eta \biggr) \\ \leq &\frac{1}{(s_{1}-t_{0})(t_{1}-t_{0})} \int _{t}^{s_{1}} \int _{t_{0}}^{t_{1}}\Phi \bigl( f^{\ast }(\xi , \eta ) \bigr) \Delta \xi \Delta \eta . \end{aligned}$$
(35)

Similarly,

$$\begin{aligned} \Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr) =&\Psi \biggl( \frac{1}{(k_{1}-t_{0})(r_{1}-t_{0})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}g^{ \ast }(\xi , \eta )\Delta \xi \Delta \eta \biggr) \\ \leq &\frac{1}{(k_{1}-t_{0})(r_{1}-t_{0})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}\Psi \bigl( g^{\ast }(\xi , \eta ) \bigr) \Delta \xi \Delta \eta . \end{aligned}$$
(36)

By multiplying (35) and (36), we get

$$\begin{aligned} &\Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr)\Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr) \\ &\quad \leq \frac{1}{(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0})} \\ &\qquad {}\times \biggl( \int _{t}^{s_{1}} \int _{t_{0}}^{t_{1}}\Phi \bigl( f^{ \ast }(\xi , \eta ) \bigr) \Delta \xi \Delta \eta \biggr) \\ &\qquad {}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}\Psi \bigl( g^{\ast }(\xi , \eta ) \bigr) \Delta \xi \Delta \eta \biggr) . \end{aligned}$$

This implies that

$$\begin{aligned} &\Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr)\Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr) (s_{1}-t_{0}) (t_{1}-t_{0}) (k_{1}-t_{0}) (r_{1}-t_{0}) \\ &\quad \leq \biggl( \int _{t}^{s_{1}} \int _{t_{0}}^{t_{1}}\Phi \bigl( f^{ \ast }(\xi ,\eta ) \bigr) \Delta \xi \Delta \eta \biggr) \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}\Psi \bigl( g^{\ast }(\xi , \eta ) \bigr) \Delta \xi \Delta \eta \biggr) . \end{aligned}$$
(37)

By Hölder’s inequality (8), we find

$$\begin{aligned} &\Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr)\Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr) (s_{1}-t_{0}) (t_{1}-t_{0}) (k_{1}-t_{0}) (r_{1}-t_{0}) \\ &\quad \leq \bigl[ (s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] ^{ \frac{p_{1}-1}{p_{1}}} \bigl[ (k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] ^{\frac{q_{1}-1}{q_{1}}} \\ &\qquad {}\times \biggl( \int _{t}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ \Phi \bigl(f^{ \ast }( \xi , \eta )\bigr) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ \Psi \bigl(g^{\ast }( \xi , \eta )\bigr) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$

Applying Young’s inequality on the term \([ (s_{1}-t_{0})(t_{1}-t_{0}) ] ^{(p_{1}-1)/p_{1}} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{(q_{1}-1)/q_{1}}\), we get

$$\begin{aligned} &\Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr)\Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr) (s_{1}-t_{0}) (t_{1}-t_{0}) (k_{1}-t_{0}) (r_{1}-t_{0}) \\ &\quad \leq \biggl( \frac{[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}}{p_{1}}+ \frac{ [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}}{q_{1}} \biggr) \\ &\qquad {}\times \biggl( \int _{t}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ \Phi \bigl(f^{ \ast }( \xi , \eta )\bigr) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ \Psi \bigl(g^{\ast }( \xi , \eta )\bigr) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$

This implies that

$$\begin{aligned} &\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0})}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}} \\ &\quad \leq \frac{1}{p_{1}q_{1}} \biggl( \int _{t}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ \Phi \bigl(f^{\ast }( \xi , \eta )\bigr) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ \Psi \bigl(g^{\ast }( \xi , \eta )\bigr) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$
(38)

Integrating both sides of (28) first with respect to \(r_{1}\) and \(k_{1}\) and then with respect to \(s_{1}\) and \(t_{1}\), respectively, and applying Hölder’s inequality (8) with indices \(p_{1}\), \(p_{1}/(p_{1}-1)\) and \(q_{1}\), \(q_{1}/(q_{1}-1)\), we get

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0})}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}} \\ &\qquad {}\times\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq \frac{1}{p_{1}q_{1}} \bigl[ (x-t_{0}) (y-t_{0}) \bigr] ^{ \frac{p_{1}-1}{p_{1}}} \bigl[ (z-t_{0}) (w-t_{0}) \bigr] ^{\frac{q_{1}-1}{q_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ \Phi \bigl(f^{\ast }( \xi , \eta )\bigr) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) \Delta s_{1}\Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ \Psi \bigl(g^{\ast }( \xi , \eta )\bigr) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) \Delta k_{1} \Delta r_{1} \biggr) ^{\frac{1}{q_{1}}} \\ &\quad =K(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ \Phi \bigl(f^{\ast }( \xi , \eta )\bigr) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) \Delta s_{1}\Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ \Psi \bigl(g^{\ast }( \xi , \eta )\bigr) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) \Delta k_{1} \Delta r_{1} \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$
(39)

Applying Fubini’s theorem and using the fact that \(\sigma (n)\geq n\), we get

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0})}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}} \\ &\qquad {}\times\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq K(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl( \sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[ \Phi \bigl(f^{\ast }(s_{1}, t_{1})\bigr) \bigr] ^{p_{1}}\Delta s_{1}\Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (w)-r_{1}\bigr) \bigl[ \Psi \bigl(g^{\ast }(k_{1}, r_{1})\bigr) \bigr] ^{q_{1}} \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}}, \end{aligned}$$

which is (33). This completes the proof. □

By using the relations (5) and taking \(\mathbb{T}_{1}= \mathbb{T}_{2}=\mathbb{R}\), \(t_{0}=0\) in Theorem 6, we get the following result.

Corollary 5

Assume that \(f^{\ast }(s_{1}, t_{1})\), \(g^{\ast }(k_{1}, r_{1})\) are real-valued continuous functions and define

$$ F^{\ast }(s_{1}, t_{1})=\frac{1}{s_{1}t_{1}} \int _{0}^{s_{1}} \int _{0}^{t_{1}}f^{\ast }(\xi , \eta )\,d\xi \,d\eta ,\qquad G^{ \ast }(k_{1}, r_{1})=\frac{1}{k_{1}r_{1}} \int _{0}^{k_{1}} \int _{0}^{r_{1}}g^{\ast }(\xi , \eta )\,d\xi \,d\eta . $$

Then for \((s_{1}, t_{1})\in I_{x}\times I_{y}\) and \((k_{1}, r_{1})\in I_{z}\times I_{w}\), we have

$$\begin{aligned} & \int _{0}^{x} \int _{0}^{y} \biggl( \int _{0}^{z} \int _{0}^{w} \frac{(s_{1}t_{1})(k_{1}r_{1})\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))}{q_{1}(s_{1}t_{1})^{{p_{1}-1}}+p_{1}(kr)^{{q-1}}}\,dk_{1} \,dr_{1} \biggr) \,ds_{1}\,dt_{1} \\ &\quad \leq K^{\ast }(p_{1}, q_{1}) \biggl( \int _{0}^{x} \int _{0}^{y}(x-s_{1}) (y-t_{1}) \bigl[ \Phi \bigl(f^{\ast }(s_{1}, t_{1})\bigr) \bigr] ^{p_{1}}\,ds_{1} \,dt_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{0}^{z} \int _{0}^{w}(z-k_{1}) (w-r_{1}) \bigl[ \Psi \bigl(g^{\ast }(k_{1}, r_{1})\bigr) \bigr] ^{q_{1}}\,dk_{1} \,dr_{1} \biggr) ^{\frac{1}{q_{1}}}, \end{aligned}$$

where

$$ K^{\ast }(p_{1}, q_{1})=\frac{1}{p_{1}q_{1}}(xy)^{ \frac{p_{1}-1}{p_{1}}}(zw)^{\frac{q_{1}-1}{q_{1}}}. $$

By using the relations (5) and taking \(\mathbb{T}_{1}= \mathbb{T}_{2}=\mathbb{Z}\), \(t_{0}=0\) in Theorem 6, we get the following result.

Corollary 6

Assume that \(\{a_{m_{1}, n_{1}}\}_{0\leq m_{1}, n_{1}\leq N}\), \(\{b_{k_{1}, r_{1}}\}_{0\leq k_{1}, r_{1}\leq N}\) are two nonnegative sequences of real numbers and define

$$ A_{{m_{1}, n_{1}}}=\frac{1}{m_{1}n_{1}}\sum_{\xi =1}^{m_{1}} \sum_{\eta =1}^{n_{1}}a_{\xi , \eta },\qquad B_{{k_{1},r_{1}}}=\frac{1}{k_{1}r_{1}}\sum_{\xi =1}^{k_{1}} \sum_{\eta =1}^{r_{1}}b_{ \xi , \eta }. $$

Then

$$\begin{aligned} &\sum_{s_{1}=1}^{m_{1}}\sum _{t_{1}=1}^{n_{1}} \Biggl( \sum _{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{w_{1}} \frac{(s_{1}t_{1})(k_{1}r_{1})\Phi (A_{{s_{1}, t_{1}}})\Psi (B_{k_{1}, r_{1}})}{q_{1}(s_{1}t_{1})^{{p_{1}-1}}+p_{1}(k_{1}r_{1})^{{q_{1}-1}}} \Biggr) \\ &\quad \leq K^{\ast \ast }(p_{1}, q_{1}) \Biggl\{ \sum _{s_{1}=1}^{m_{1}} \sum _{t_{1}=1}^{n_{1}}(m_{1}-s_{1}+1) (n_{1}-t_{1}+1) \bigl(\Phi (a_{s_{1}, t_{1}}) \bigr)^{p_{1}} \Biggr\} ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \Biggl\{ \sum_{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{w_{1}}(z_{1}-k_{1}+1) (w_{1}-r_{1}+1) \bigl(\Psi (b_{{k_{1}, r_{1}}}) \bigr)^{q_{1}} \Biggr\} ^{\frac{1}{q_{1}}}, \end{aligned}$$

where

$$ K^{\ast \ast }(p_{1}, q_{1})=\frac{1}{p_{1}q_{1}}(m_{1}n_{1})^{ \frac{p_{1}-1}{p_{1}}}(z_{1}w_{1})^{\frac{q_{1}-1}{q_{1}}}. $$

Theorem 7

Let (H1), (H2), and (H4) be satisfied, \(f^{\ast }(s_{1}, t_{1})\in \mathrm{CC}_{rd}^{1}(I_{x}\times I_{y}, \mathbb{R}^{+})\), \(g^{\ast }(k_{1}, r_{1})\in \mathrm{CC}_{rd}^{1}(I_{z}\times I_{w}, \mathbb{R}^{+})\), and \(p^{\ast }(\xi , \eta )\), \(q^{\ast }(\xi , \eta ) \) be two positive functions. Suppose that \(P^{\ast }\) and \(Q^{\ast } \) are as defined in Theorem 5and let

$$\begin{aligned} &F^{\ast }(s_{1}, t_{1}) =\frac{1}{P^{\ast }(s_{1}, t_{1})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}}p^{\ast }(\xi , \eta )f^{ \ast }(\xi , \eta )\Delta \xi \Delta \eta , \\ &G^{\ast }(k_{1}, r_{1}) =\frac{1}{Q^{\ast }(k_{1}, r_{1})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}q^{\ast }(\xi , \eta )g^{ \ast }(\xi , \eta )\Delta \xi \Delta \eta . \end{aligned}$$
(40)

Then for \((s_{1}, t_{1})\in I_{x}\times I_{y}\) and \((k_{1}, r_{1})\in I_{z}\times I_{w}\), we have

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))P^{\ast }(s_{1}, t_{1})Q^{\ast }(k_{1}, r_{1})}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq H(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl( \sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[ P^{\ast }(s_{1}, t_{1}) \Phi \bigl(f^{\ast }(s_{1}, t_{1}) \bigr) \bigr] ^{p_{1}}\Delta s_{1} \Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (w)-r_{1}\bigr) \bigl[ Q^{\ast }(k_{1}, r_{1})\Psi \bigl(g^{\ast }(k_{1}, r_{1}) \bigr) \bigr] ^{q_{1}}\Delta k_{1}\Delta r_{1} \biggr) ^{ \frac{1}{q_{1}}}, \end{aligned}$$
(41)

where

$$ H(p_{1}, q_{1})=\frac{1}{p_{1}q_{1}}\bigl[(s_{1}-t_{0}) (t_{1}-t_{0})\bigr]^{\frac{p_{1}-1}{p_{1}}}\bigl[(k_{1}-t_{0}) (r_{1}-t_{0})\bigr]^{ \frac{q_{1}-1}{q_{1}}}. $$
(42)

Proof

By assumption and using the Jensen’s inequality (9), it follows that

$$\begin{aligned} \Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr) =&\Phi \biggl( \frac{1}{P^{\ast }(s_{1}, t_{1})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}}p^{ \ast }(\xi , \eta )f^{\ast }(\xi , \eta )\Delta \xi \Delta \eta \biggr) \\ \leq &\frac{1}{P^{\ast }(s_{1}, t_{1})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}}p^{\ast }(\xi , \eta )\Phi \bigl(f^{\ast }(\xi , \eta )\bigr)\Delta \xi \Delta \eta \end{aligned}$$
(43)

and

$$\begin{aligned} \Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr) =&\Psi \biggl( \frac{1}{Q^{\ast }(k_{1}, r_{1})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}q^{ \ast }(\xi , \eta )g^{\ast }(\xi , \eta )\Delta \xi \Delta \eta \biggr) \\ \leq &\frac{1}{Q^{\ast }(k_{1}, r_{1})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}q^{\ast }(\xi , \eta )\Psi \bigl(g^{\ast }(\xi , \eta )\bigr)\Delta \xi \Delta \eta . \end{aligned}$$
(44)

From (43) and (44) and using Hölder’s inequality (8) with \(p_{1}\), \(p_{1}/(p_{1}-1)\) and \(q_{1}\), \(q_{1}/(q_{1}-1)\), respectively, we get

$$\begin{aligned} \Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr) \leq & \frac{1}{P^{\ast }(s_{1}, t_{1})} \bigl[ (s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] ^{ \frac{p_{1}-1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ p^{ \ast }(\xi , \eta ) \Phi \bigl(f^{\ast }(\xi , \eta )\bigr) \bigr] ^{p_{1}} \Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \end{aligned}$$
(45)

and

$$\begin{aligned} \Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr) \leq & \frac{1}{Q^{\ast }(k_{1}, r_{1})} \bigl[ (k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] ^{ \frac{q_{1}-1}{q_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ q^{ \ast }(\xi , \eta ) \Psi \bigl(g^{\ast }(\xi , \eta )\bigr) \bigr] ^{q_{1}} \Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$
(46)

From (45) and (46) and using the elementary inequality (10), we get

$$\begin{aligned} &\Phi \bigl(F^{\ast }(s_{1}, t_{1})\bigr)\Psi \bigl(G^{\ast }(k_{1}, r_{1})\bigr)P^{\ast }(s_{1}, t_{1})Q^{\ast }(k_{1}, r_{1}) \\ &\quad \leq \biggl( \frac{[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}}{p_{1}}+ \frac{ [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}}{q_{1}} \biggr) \\ &\qquad {}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ p^{ \ast }(\xi , \eta ) \Phi \bigl(f^{\ast }(\xi , \eta )\bigr) \bigr] ^{p_{1}} \Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ q^{ \ast }(\xi , \eta ) \Psi \bigl(g^{\ast }(\xi , \eta )\bigr) \bigr] ^{q_{1}} \Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$
(47)

This implies that

$$\begin{aligned} &\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))P^{\ast }(s_{1}, t_{1})Q^{\ast }(k_{1}, r_{1})}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}} \\ &\quad \leq \frac{1}{p_{1}q_{1}} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ p^{\ast }(\xi , \eta ) \Phi \bigl(f^{\ast }(\xi , \eta )\bigr) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ q^{ \ast }(\xi , \eta ) \Psi \bigl(g^{\ast }(\xi , \eta )\bigr) \bigr] ^{q_{1}} \Delta \xi \Delta \eta \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$
(48)

Integrating both sides of (48) with respect to \(r_{1}\) and \(k_{1}\) and then with respect to \(s_{1}\) and \(t_{1}\), respectively, and applying Hölder’s inequality (8) with indices \(p_{1}\), \(p_{1}/(p_{1}-1)\) and \(q_{1}\), \(q_{1}/(q_{1}-1)\), we see that

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))P^{\ast }(s_{1}, t_{1})Q^{\ast }(k_{1}, r_{1})}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq \frac{1}{p_{1}q_{1}} \bigl[ (x-t_{0}) (y-t_{0}) \bigr] ^{ \frac{p_{1}-1}{p_{1}}} \bigl[ (z-t_{0}) (w-t_{0}) \bigr] ^{\frac{q_{1}-1}{q_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ p^{\ast }(\xi , \eta ) \Phi \bigl(f^{ \ast }(\xi , \eta )\bigr) \bigr] ^{p_{1}}\Delta \xi \Delta \eta \biggr) \Delta s_{1}\Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ q^{\ast }(\xi , \eta ) \Psi \bigl(g^{ \ast }(\xi , \eta )\bigr) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}} \\ &\quad =H(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[ p^{\ast }( \xi , \eta ) \Phi \bigl(f^{\ast }(\xi , \eta )\bigr) \bigr] ^{p_{1}} \Delta \xi \Delta \eta \biggr) \Delta s_{1}\Delta t_{1} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[ q^{\ast }(\xi , \eta ) \Psi \bigl(g^{ \ast }(\xi , \eta )\bigr) \bigr] ^{q_{1}}\Delta \xi \Delta \eta \biggr) \Delta k_{1}\Delta r_{1} \biggr) ^{\frac{1}{q_{1}}}. \end{aligned}$$
(49)

Applying Fubini’s theorem and using the fact that \(\sigma (n)\geq n\), we get

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))P^{\ast }(s_{1}, t_{1})Q^{\ast }(k_{1}, r_{1})}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq H(p_{1}, q_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl( \sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[ p^{\ast }(s_{1}, t_{1}) \Phi \bigl(f^{\ast }(s_{1}, t_{1}) \bigr) \bigr] ^{p_{1}}\Delta s_{1} \Delta t_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (w)-r_{1}\bigr) \bigl[ q^{\ast }(k_{1}, r_{1})\Psi \bigl(g^{\ast }(k_{1}, r_{1}) \bigr) \bigr] ^{q_{1}}\Delta k_{1}\Delta r_{1} \biggr) ^{ \frac{1}{q_{1}}}, \end{aligned}$$

which is (41). This completes the proof. □

By using the relations (5) and taking \(\mathbb{T}_{1}= \mathbb{T}_{2}=\mathbb{R}\), \(t_{0}=0\) in Theorem 7, we get the following result.

Corollary 7

Assume that \(f^{\ast }(s_{1}, t_{1})\), \(g^{\ast }(k_{1}, r_{1})\) are real-valued continuous functions, \(p^{\ast }(s_{1}, t_{1})\), \(q^{\ast }(k_{1}, r_{1})\) are two positive functions, and define

$$\begin{aligned} &F^{\ast }(s_{1}, t_{1}) =\frac{1}{P^{\ast }(s_{1}, t_{1})} \int _{0}^{s_{1}} \int _{0}^{t_{1}}p^{\ast }(\xi , \eta )f^{ \ast }(\xi ,\eta )\,d\xi \,d\eta ,\\ & P^{\ast }(s_{1}, t_{1})= \int _{0}^{s_{1}} \int _{0}^{t_{1}}p^{\ast }(\xi , \eta )\,d \xi \,d\eta, \\ &G^{\ast }(k_{1}, r_{1}) =\frac{1}{Q^{\ast }(k_{1}, r_{1})} \int _{0}^{k_{1}} \int _{0}^{r_{1}}q^{\ast }(\xi ,\eta )g^{\ast }(\xi , \eta )\,d\xi \,d\eta ,\\ & Q^{\ast }(k_{1}, r_{1})= \int _{0}^{k_{1}} \int _{0}^{r_{1}}q^{\ast }(\xi , \eta )\,d\xi \,d\eta . \end{aligned}$$

Then for \((s_{1}, t_{1})\in I_{x}\times I_{y}\) and \((k_{1}, r_{1})\in I_{z}\times I_{w}\), we get

$$\begin{aligned} & \int _{0}^{x} \int _{0}^{y} \biggl( \int _{0}^{z} \int _{0}^{w} \frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))P^{\ast }(s_{1},t_{1})Q^{\ast }(k_{1}, r_{1})}{q_{1}(s_{1}t_{1})^{{p_{1}-1}}+p_{1}(k_{1}r_{1})^{{q_{1}-1}}}\,dk_{1} \,dr_{1} \biggr) \,ds_{1}\,dt_{1} \\ &\quad \leq H^{\ast }(p_{1}, q_{1}) \biggl( \int _{0}^{x} \int _{0}^{y}(x-s_{1}) (y-t_{1}) \bigl[ p^{\ast }(s_{1}, t_{1}) \Phi \bigl(f^{\ast }(s_{1}, t_{1})\bigr) \bigr] ^{p_{1}}\,ds_{1}\,dt_{1} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad {}\times \biggl( \int _{0}^{z} \int _{0}^{w}(z-k_{1}) (w-r_{1}) \bigl[ q^{ \ast }(k_{1}, r_{1}) \Psi \bigl(g^{\ast }(k_{1}, r_{1})\bigr) \bigr] ^{q_{1}}\,dk_{1}\,dr_{1} \biggr) ^{\frac{1}{q_{1}}}, \end{aligned}$$

where

$$ H^{\ast }(p_{1}, q_{1})=\frac{1}{p_{1}q_{1}}(xy)^{ \frac{p_{1}-1}{p_{1}}}(zw)^{\frac{q_{1}-1}{q_{1}}}. $$

By using the relations (5) and taking \(\mathbb{T}_{1}= \mathbb{T}_{2}=\mathbb{Z}\), \(t_{0}=0\) in Theorem 7, we get the following result.

Corollary 8

Assume that \(\{a_{m_{1}, n_{1}}\}_{0\leq m_{1}, n_{1}\leq N}\), \(\{b_{k_{1}, r_{1}}\}_{0\leq k_{1}, r_{1}\leq N}\) be two nonnegative sequences of real numbers and \(\{p_{m_{1}, n_{1}}\}_{{0\leq m_{1}, n_{1}\leq N}}\), \(\{q_{k_{1}, r_{1}}\}_{0\leq k_{1}, r_{1}\leq N}\) be positive sequences and define

$$\begin{aligned} &A_{{m_{1}, n_{1}}} =\frac{1}{P_{m_{1}, n_{1}}}\sum_{ \xi =1}^{m_{1}} \sum_{\eta =1}^{n_{1}}a_{\xi , \eta },\qquad P_{m_{1},n_{1}}=\sum_{\xi =1}^{m_{1}}\sum _{\eta =1}^{n_{1}}p_{\xi , \eta }, \\ &B_{{k_{1}, r_{1}}} =\frac{1}{Q_{{k_{1}, r_{1}}}} \sum_{\xi =1}^{k_{1}} \sum_{\eta =1}^{r_{1}}b_{\xi , \eta },\qquad Q_{{k_{1},r_{1}}}=\sum_{\xi =1}^{k_{1}}\sum _{\eta =1}^{r_{1}}q_{\xi , \eta }. \end{aligned}$$

Then

$$\begin{aligned} &\sum_{s_{1}=1}^{m_{1}}\sum _{t_{1}=1}^{n_{1}} \Biggl( \sum _{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{w_{1}} \frac{\Phi (A_{{s_{1}, t_{1}}})\Psi (B_{k_{1}, r_{1}})P_{s_{1}, t_{1}}Q_{{k_{1},r_{1}}}}{q_{1}(s_{1}t_{1})^{{p_{1}-1}}+p_{1}(k_{1}r_{1})^{{q_{1}-1}}} \Biggr) \\ &\quad \leq H^{\ast \ast }(p_{1}, q_{1}) \Biggl( \sum _{s_{1}=1}^{m_{1}}\sum _{t_{1}=1}^{n_{1}}(m_{1}-s_{1}+1) (n_{1}-t_{1}+1)\bigl[p_{s_{1}, t_{1}}\Phi (a_{s_{1}, t_{1}})\bigr]^{p_{1}} \Biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad {}\times \Biggl( \sum_{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{w_{1}}(z_{1}-k_{1}+1) (w_{1}-r_{1}+1)\bigl[q_{k_{1}, r_{1}}\Psi (b_{{k_{1}, r_{1}}})\bigr]^{q_{1}} \Biggr) ^{ \frac{1}{q_{1}}}, \end{aligned}$$

where

$$ H^{\ast \ast }(p_{1}, q_{1})=\frac{1}{p_{1}q_{1}}(m_{1}n_{1})^{ \frac{p_{1}-1}{p_{1}}}(z_{1}w_{1})^{\frac{q_{1}-1}{q_{1}}}. $$

Remark 2

By applying (10) on (33) in Theorem 6 and (41) in Theorem 7, respectively, we get the following inequalities:

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0})}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}} \\ &\qquad {}\times\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq K(p_{1}, q_{1}) \biggl\{ \frac{1}{p_{1}} \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[ \Phi \bigl(f^{ \ast }(s_{1}, t_{1})\bigr) \bigr] ^{p_{1}}\Delta s_{1}\Delta t_{1} \biggr) \\ &\qquad {} +\frac{1}{q_{1}} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl( \sigma (z)-k_{1} \bigr) \bigl(\sigma (w)-r_{1}\bigr) \bigl[ \Psi \bigl(g^{\ast }(k_{1}, r_{1})\bigr) \bigr] ^{q_{1}}\Delta k_{1}\Delta r_{1} \biggr) \biggr\} , \end{aligned}$$

where \(K(p_{1}, q_{1})\) is defined in (34), and

$$\begin{aligned} & \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\frac{\Phi (F^{\ast }(s_{1}, t_{1}))\Psi (G^{\ast }(k_{1}, r_{1}))P^{\ast }(s_{1}, t_{1})Q^{\ast }(k_{1}, r_{1})}{q_{1}[(s_{1}-t_{0})(t_{1}-t_{0})]^{{p_{1}-1}}+p_{1} [ (k_{1}-t_{0})(r_{1}-t_{0}) ] ^{q_{1}-1}}\Delta k_{1}\Delta r_{1} \biggr) \Delta s_{1}\Delta t_{1} \\ &\quad \leq H(p_{1}, q_{1}) \biggl\{ \frac{1}{p_{1}} \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y}\bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[ p^{\ast }(s_{1}, t_{1})\Phi \bigl(f^{\ast }(s_{1}, t_{1}) \bigr) \bigr] ^{p_{1}} \Delta s_{1}\Delta t_{1} \biggr) \\ &\qquad {} +\frac{1}{q_{1}} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{w}\bigl( \sigma (z)-k_{1} \bigr) \bigl(\sigma (w)-r_{1}\bigr) \bigl[ q^{\ast }(k_{1}, r_{1}) \Psi \bigl(g^{\ast }(k_{1}, r_{1}) \bigr) \bigr] ^{q_{1}}\Delta k_{1} \Delta r_{1} \biggr) \biggr\} , \end{aligned}$$

where \(H(p_{1}, q_{1})\) is defined in (42).

Remark 3

Clearly, for the one-dimensional case, Theorems 4, 5, 6, and 7, coincide with Corollary 3.3, Theorems 3.2, 3.3, and 3.4, respectively, of [7].

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Acknowledgements

This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

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This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

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Abd El-Hamid, H.A., Rezk, H.M., Ahmed, A.M. et al. Some dynamic Hilbert-type inequalities for two variables on time scales. J Inequal Appl 2021, 31 (2021). https://doi.org/10.1186/s13660-021-02559-1

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MSC

  • 26D15
  • 34A40
  • 39A12
  • 34N05

Keywords

  • Hilbert’s inequality
  • Young’s inequality
  • Hölder’s inequality
  • Jensen’s inequality
  • Weights
  • Kernel
  • Time scales