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New dynamic Hilbert-type inequalities in two independent variables involving Fenchel–Legendre transform

Abstract

In this paper, we establish some dynamic Hilbert-type inequalities in two independent variables on time scales by using the Fenchel–Legendre transform. We also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as particular cases. Our results give more general forms of several previously established inequalities.

Introduction

Hilger [1] presented a new calculus, named time scales, to get a unification of discrete and continuous calculus. The monographs by Bohner and Peterson [2, 3] summarize and organize much of time scales calculus.

A time scale \(\mathbb{T}\) is an arbitrary nonempty closed subset of the real numbers. We assume that \(\mathbb{T}\) has the topology inherited from the standard topology on the real numbers \(\mathbb{R}\). We define the forward jump operator \(\sigma : \mathbb{T}\rightarrow \mathbb{T}\) for any \(t\in \mathbb {T}\) by

$$ \sigma (t):=\inf \{s\in \mathbb{T} : s>t\} $$

and the backward jump operator \(\rho : \mathbb {T}\to \mathbb {T}\) by

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

In the previous two definitions, we set \(\inf \emptyset =\sup \mathbb {T}\) (i.e., if t is the maximum of \(\mathbb {T}\), then \(\sigma (t)=t\)) and \(\sup \emptyset =\inf \mathbb {T}\) (i.e., if t is the minimum of \(\mathbb {T}\), then \(\rho (t)=t\)), where is the empty set.

A point \(t\in \mathbb {T}\) is said to be right-scattered if \(\sigma (t)>t\), right-dense if \(\sigma (t)=t\) and \(t<\sup \mathbb {T}\), left-scattered if \(\rho (t)< t\), and left-dense if \(\rho (t)=t\) and \(t > \inf \mathbb {T}\). The points that are right- and left-scattered are called isolated. The points that are right- and left-dense are called dense.

We define the forward graininess function \(\mu :\mathbb {T}\to [0,\infty )\) by \(\mu (t):= \sigma (t)-t\) for \(t \in \mathbb {T}\).

Let \(f : \mathbb {T}\to \mathbb {R}\). Then the function \(f^{\sigma } : \mathbb {T}\to \mathbb {R}\) is defined by \(f^{\sigma }(t)=f(\sigma (t))\) for \(t\in \mathbb {T}\), that is, \(f^{\sigma }=f \circ \sigma \). Similarly, the function \(f^{\rho } : \mathbb {T}\to \mathbb {R}\) is defined by \(f^{\rho }(t)=f(\rho (t))\) for \(t\in \mathbb {T}\), that is, \(f^{\rho }=f\circ \rho \).

We introduce the set \(\mathbb {T}^{\kappa }\) as follows: If \(\mathbb {T}\) has a left-scattered maximum m, then \(\mathbb {T}^{\kappa }=\mathbb {T}-\{m\}\), otherwise, \(\mathbb {T}^{\kappa } = \mathbb {T}\).

The interval \([a,b]\) in \(\mathbb {T}\) is defined by

$$ [a,b]_{\mathbb {T}}=\{t\in \mathbb {T}:a\leq t\leq b\}. $$

Open intervals and half-closed intervals are defined similarly.

Let \(f : \mathbb{T} \rightarrow \mathbb{R}\) be a real-valued function on a time scale \(\mathbb{T}\). Then, for \(t\in \mathbb{T}^{\kappa }\), we define \(f^{\Delta }(t)\) to be the number (if it exists) such that for any \(\varepsilon > 0\), there is a neighborhood U of t such that, for all \(s\in U\), we have

$$ \bigl\vert \bigl[f\bigl(\sigma (t)\bigr)-f(s)\bigr]-f^{\Delta }(t)\bigl[ \sigma (t)-s\bigr] \bigr\vert \leq \varepsilon \bigl\vert \sigma (t)-s \bigr\vert . $$

In this case, we say that f is delta-differentiable on \(\mathbb{T}^{\kappa } \), provided that \(f^{\Delta }(t)\) exists for all \(t\in \mathbb{T}^{\kappa }\).

Now we recall some basic concepts related to partial derivatives on time scales. Let \(\mathbb{T}_{1}\) and \(\mathbb{T}_{2}\) be any time scales. Let \(\sigma _{1}\) and \(\Delta _{1}\) (\(\sigma _{2}\) and \(\Delta _{2}\)) denote the forward jump operator and delta differentiation operator on \(\mathbb{T}_{1}\) (resp., \(\mathbb{T}_{2}\)). Assume that \(u < v\) are points in \(\mathbb{T}_{1}\), \(e < f\) are points in \(\mathbb{T}_{2}\), \([u,v)\) 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 \(\mathbb{T}_{1} \times \mathbb{T}_{2}\),

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

Let \(f:\mathbb{T}_{1} \times \mathbb{T}_{2} \longrightarrow \mathbb{R}\). We say that f has a \(\Delta _{1}\) partial derivative at \((t _{1},t _{2}) \in \mathbb{T}_{1} \times \mathbb{T}_{2} \) with respect to \(t _{1}\) if for each \(\epsilon > 0\), there exists a neighborhood \(U_{t _{1}}\) of \(t _{1}\) such that

$$ \bigl\vert \bigl[f\bigl(\sigma _{1}(t _{1}),t _{2}\bigr)- f(s ,t _{2})\bigr]-f^{\Delta _{1}}(t _{1},t _{2})\bigl[\sigma _{1}(t _{1})-s \bigr] \bigr\vert \leqslant \epsilon \bigl\vert \sigma _{1}(t _{1})-s \bigr\vert $$

for all \(s \in U_{t _{1}}\). We say that f has a \(\Delta _{2}\) partial derivative at \((t _{1},t _{2}) \in \mathbb{T}_{1} \times \mathbb{T}_{2} \) with respect to \(t _{2}\) if for each \(\epsilon > 0\), there exists a neighborhood \(U_{t _{2}}\) of \(t _{2}\) such that

$$ \bigl\vert \bigl[f(t _{1},\bigl(\sigma _{2}(t _{2})\bigr)- f(t _{1},l )\bigr]-f^{\Delta _{2}}(t _{1},t _{2})\bigl[\sigma _{2}(t _{2})-l \bigr] \bigr\vert \leqslant \epsilon \bigl\vert \sigma _{2}(t _{2})-l \bigr\vert $$

for all \(l \in U_{t _{2}}\).

A function \(f:\mathbb{T}_{1} \times \mathbb{T}_{2} \longrightarrow \mathbb{R}\) is said to be rd-continuous in \(t _{2}\) if for every \(\beta _{1}\in \mathbb{T}_{1}\), the function \(f(\beta _{1},t _{2})\) is rd-continuous on \(\mathbb{T}_{2}\), and it is rd-continuous in \(t _{1}\) if for every \(\beta _{2}\in \mathbb{T}_{2}\), the function \(f(t _{1},\beta _{2})\) is rd-continuous on \(\mathbb{T}_{1}\).

Let \(CC_{\mathrm{rd}}\) denote the set of functions \(f(t _{1},t _{2})\) on \(\mathbb{T}_{1} \times \mathbb{T}_{2}\) with the following properties:

  1. 1.

    f is rd-continuous in \(t _{1}\).

  2. 2.

    f is rd-continuous in \(t _{2}\).

  3. 3.

    If \((x _{1},x _{2})\in \mathbb{T}_{1} \times \mathbb{T}_{2}\) with \(x _{1}\) right-dense or maximal and \(x _{2}\) right-dense or maximal, then f is continuous at \((x _{1},x _{2})\).

  4. 4.

    If \(x _{1}\) and \(x _{2}\) are both left-dense, then the limit of \(f(t _{1},t _{2})\) exists as \((t _{1},t _{2})\) approaches \((x _{1},x _{2})\) along any path in the region

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

Next, we write Hölder’s inequality and Jensen’s inequality in two independent variables on time scales.

Lemma 1.1

(Dynamic Hölder’s inequality [4])

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

$$\begin{aligned} \int _{u }^{v } \int _{u }^{v } \bigl\vert f (r ,t )g (r ,t ) \bigr\vert \Delta r \Delta t \leq & \biggl[ \int _{u }^{v } \int _{u }^{v } \bigl\vert f (r ,t ) \bigr\vert ^{p} \Delta r \Delta t \biggr]^{\frac{1}{p}} \\ &{} \times \biggl[ \int _{u }^{v } \int _{u }^{v } \bigl\vert g (r ,t ) \bigr\vert ^{q} \Delta r \Delta t \biggr]^{\frac{1}{q}}. \end{aligned}$$
(1.1)

Lemma 1.2

(Dynamic Jensen’s inequality [5])

Let r, t in the rectangle R, and let \(- \infty \leqslant m\), \(n \leqslant \infty \). If \(f\in CC_{\mathrm{rd}}(\mathbb{R},( m ,n ))\) and \(\Phi :( m ,n )\longrightarrow \mathbb{R}\) is convex, then

$$ \Phi \biggl( \frac{\int _{u}^{v} \int _{\omega }^{s} f (r ,t ) \Delta _{1} r \Delta _{2} t }{\int _{u}^{v} \int _{\omega }^{s}\Delta _{1} r \Delta _{2} t } \biggr) \leqslant \frac{ \int _{u}^{v} \int _{\omega }^{s} \Phi (f (r ,t ))\Delta _{1} r \Delta _{2} t }{\int _{u}^{v} \int _{\omega }^{s} \Delta _{1} r \Delta _{2} t }. $$
(1.2)

Lemma 1.3

(Fubini’s thoerem [6])

Let \((\Omega _{1} , \Sigma _{1}, \mu _{\Delta })\) and \((\Omega _{2} , \Sigma _{2}, \nu _{\Delta })\) be two finite-dimensional time scales measure spaces. Further, let \(f:\Omega _{1} \times \Omega _{2} \rightarrow \mathbb{R}\) be a delta-integrable function. Define the functions

$$ \phi (y )= \int _{\Omega _{1} }f(x ,y )\,d\mu _{\Delta }(x ),\quad y \in \Omega _{2} , $$

and

$$ \psi (x )= \int _{\Omega _{2} }f(x ,y )\,d\nu _{\Delta }(y ),\quad x \in \Omega _{1} . $$

Then ϕ is delta-integrable on \(\Omega _{2} \), ψ is delta-integrable on \(\Omega _{1} \), and

$$ \int _{X}d\mu _{\Delta }(x ) \int _{Y}f(x ,y )\,d\nu _{\Delta }(y )= \int _{Y}d \nu _{\Delta }(y ) \int _{X}f(x ,y )\,d\mu _{\Delta }(x ). $$

Lemma 1.4

([7])

Let \(y +x \geqslant 1\), where y and \(x \in \mathbb{R}\). For \(\alpha \geqslant \beta \geqslant \frac{1}{2}\) and \(\gamma >0\), we have

$$ (x +y )^{\frac{1}{\gamma }} \leqslant \bigl( \vert x \vert ^{\frac{1}{2 \beta }}+ \vert y \vert ^{\frac{1}{2 \beta }} \bigr)^{\frac{2\alpha }{\gamma }} . $$
(1.3)

Definition 1.5

A function \(\Phi : [0,\infty ) \longrightarrow \mathbb{R}\) is called a submultiplicative function if

$$ \Phi (x y )\leqslant \Phi (x ) \Phi (y )\quad \text{for all } x ,y \geqslant 0. $$
(1.4)

Now we present the Fenchel–Legendre transform, which will need in the proof of our results. We refer to example to [810] for more detail.

Definition 1.6

A function \(f: \mathbb{R}^{n} \longrightarrow \mathbb{R} \cup \{-\infty , \infty \}\) is called coercive if

$$ f(x)\longrightarrow \infty \quad \mbox{as } \Vert x \Vert \longrightarrow \infty . $$

Definition 1.7

Let \(h:{\mathbb{R}}^{n} \longrightarrow \mathbb{R} \cup \{+\infty \}\) be a function such that \(h\neq +\infty \), that is, \(\operatorname{Dom}(h)=\{ x \in \mathbb{R}^{n} |h(x)< \infty \}\neq \emptyset \). Then the Fenchel–Legendre transform is defined as

$$ h^{*}:{\mathbb{R}}^{n}\longrightarrow \mathbb{R} \cup \{+\infty \},\qquad y\longrightarrow h^{*}(y)=\sup \bigl\{ \langle y,x\rangle -h(x), x\in \operatorname{Dom}(h)\bigr\} , $$
(1.5)

where \(\langle \cdot ,\cdot \rangle \) denotes the inner product in \({\mathbb{R}}^{n}\). The mapping \(h\longrightarrow h^{*}\) is called the conjugate operation.

The domain of \(h^{*}\) is the set of slopes of all affine functions minorizing the function h over \({\mathbb{R}}^{n}\). An equivalent formula for (1.5) is obtained in the next corollary:

Corollary 1.8

Let \(h:{\mathbb{R}}^{n} \longrightarrow \mathbb{R}\) be strictly convex differentiable 1-coercive function. Then

$$ h^{*}(y)= \bigl\langle y,(\nabla h)^{-1}(y)\bigr\rangle -h \bigl((\nabla h)^{-1}(y)\bigr) $$
(1.6)

for all \(y\in \operatorname{Dom}(h^{*})\), where \(\langle \cdot ,\cdot \rangle \) denotes the inner product in \({\mathbb{R}}^{n}\).

Lemma 1.9

(Fenchel–Young inequality [10])

Let h and \(h^{*}\) be a function and its Fenchel–Legendre transform, respectively. Then

$$ \langle x ,y \rangle \leqslant h(x )+h^{*}(y ) $$
(1.7)

for all \(y \in \operatorname{Dom}(h^{*})\) and \(x \in \operatorname{Dom}(h)\).

It is well known that Hilbert’s inequalities play a very important role in the mathematical analysis. Several dynamic inequalities of Hilbert type and others are established by different mathematicians; see [4, 948].

Hamiaz et al. [20] studied the following discrete inequalities: Suppose \(q, p\geqslant 1\), \(\alpha \geqslant \beta \geqslant \frac{1}{2}\), and \((b_{m})_{m}\geq 0\) and \((a_{n})_{n}\geq 0\) are sequences of real numbers. Denote \(A_{n}=\sum_{s=1}^{n} a_{s}\) and \(B_{m}=\sum_{t=1}^{m} b_{t}\). Then

$$\begin{aligned} \sum_{n=1}^{k} \sum _{m=1}^{r} \frac{A^{2p}_{n} B^{2q}_{m}}{h(n)+h^{*}(m)} \leqslant& C _{1}(p,q) \Biggl(\sum_{n=1}^{k}(k-n+1) \bigl(a_{n} A^{p-1}_{n}\bigr)^{2} \Biggr) \\ &{}\times \Biggl(\sum_{m=1}^{r}(r-m+1) \bigl(b_{m} B^{q-1}_{m}\bigr)^{2} \Biggr) \end{aligned}$$

and

$$\begin{aligned} \sum_{n=1}^{k} \sum _{m=1}^{r} \frac{A^{p}_{n} B^{q}_{m}}{ ( \vert h(n) \vert ^{\frac{1}{2\beta }} + \vert h^{*}(m) \vert ^{\frac{1}{2 \beta }} )^{\alpha }} \leqslant& \sum_{n=1}^{k} \sum _{m=1}^{r} \frac{A^{p}_{n} B^{q}_{m}}{\sqrt{h(n)+h^{*}(m)}} \\ \leqslant& C _{2}(p,q,k,r) \Biggl(\sum _{n=1}^{k}(k-n+1) \bigl(a_{n} A^{p-1}_{n}\bigr)^{2} \Biggr)^{\frac{1}{2}} \\ &{} \times \Biggl(\sum_{m=1}^{r}(r-m+1) \bigl(b_{m} B^{q-1}_{m}\bigr)^{2} \Biggr)^{\frac{1}{2}} \end{aligned}$$

unless \((a_{n})\) or \((b_{m})\) is null, where

$$ C _{1}(p,q)=(pq)^{2} \quad \text{and}\quad C _{2}(p,q,r,k)=pq \sqrt{kr}. $$

Very recently, El-Deeb et al. [7] studied the time scales version of the above inequalities. They proved that if \(A(s ):= \int _{t_{0}}^{s } a(\tau ) \Delta \tau \) and \(B(t ):= \int _{t_{0}}^{t } b(\tau ) \Delta \tau \), where \(a(\tau )\geq 0\) and \(b(\tau )\geq 0\) are right-dense continuous, then

$$\begin{aligned}& \int _{t_{0}}^{x } \int _{t_{0}}^{y } \frac{A^{qK}(\sigma (s ))B^{qL}(\sigma (t ))}{ ( \vert h(\sigma (s )-t_{0}) \vert ^{\frac{1}{2\beta }}+ \vert h^{*}(\sigma (t )-t_{0}) \vert ^{\frac{1}{2\beta }} )^{\frac{2q \alpha }{p}}} \Delta s \Delta t \\& \quad \leqslant C^{**}_{1}(L,K,q) \biggl( \int _{t_{0}}^{x } \bigl(\sigma ({x })- \sigma (s ) \bigr) \bigl( a(s )A^{K-1}\bigl(\sigma (s ) \bigr)\bigl)^{q} \Delta s \biggl) \\& \qquad {}\times \biggl( \int _{t_{0}}^{{y }} \bigl(\sigma ({y })-\sigma (t ) \bigr) \bigl( b(t )B^{L-1}\bigl(\sigma (t ) \bigr)\bigl)^{q} \Delta t \biggl) \end{aligned}$$

and

$$\begin{aligned}& \int _{t_{0}}^{x } \int _{t_{0}}^{y } \frac{A^{K}(\sigma (s ))B^{L}(\sigma (t ))}{ ({ \vert h(\sigma (s )-t_{0}) \vert }^{\frac{1}{2\beta }}+{ \vert h^{*}(\sigma (t )-t_{0}) \vert }^{\frac{1}{2\beta }} )^{\frac{2 \alpha }{p}}} \Delta s \Delta t \\& \quad \leqslant C^{**}_{2}(L,K,p) \biggl( \int _{t_{0}}^{x } \bigl(\sigma ({x })- \sigma (s ) \bigr) \bigl(A^{K-1}\bigl(\sigma (s )\bigr)a(s ) \bigr)^{q} \Delta s \biggr)^{\frac{1}{q}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{y } \bigl(\sigma ({y })-\sigma (t ) \bigr) { \bigl(B^{L-1}\bigl(\sigma (t )\bigr)b(t ) \bigr)^{q} \Delta t \biggr)}^{ \frac{1}{q}} \end{aligned}$$

for \(p, q > 1\), \(K, L \geqslant 1\), and \(s, t, t_{0}, {x }, {y } \in \mathbb{T}\), and

$$ C^{**}_{1}(L,K,q)= (KL)^{q}\quad \text{and}\quad C^{**}_{2}(L,K,p)=KL (x -t_{0})^{\frac{1}{p}}(y -t_{0})^{\frac{1}{p}}. $$

In this paper, motivated by the inequalities mentioned, using the Fenchel–Legendre transform and other vital tools, we prove some new Hilbert-type inequalities in two independent variables on time scales. These inequalities extend and give more general new forms of several previously established inequalities. For example, we generalize the results given in [7, 20] and others.

Main results

In the next theorems, we assume that \(\alpha \geq \beta \geq \frac{1}{2} \) and \(p_{1}, q_{1}>1\), \(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\). 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 \(I_{\omega }=[t_{0},\omega )_{{\mathbb{T}}_{2}}\), where x, \(z \in \Omega _{1}= [t_{0},\infty ) \cap \mathbb{T}_{1}\), y, \(\omega \in \Omega _{2}= [t_{0},\infty ) \cap \mathbb{T}_{2}\), and \(0\leq t_{0}< r_{1}< k_{1}< t_{1}<s_{1}\).

Theorem 2.1

Let \(\mathbb{T}_{1}\) and \(\mathbb{T}_{2}\) be any time scales with \(t_{0}\), \(s_{1}\), \(k_{1}\), x, z \(\in \mathbb{T}_{1}\) and \(t_{0}\), \(t_{1}\), \(r_{1}\), y, ω \(\in \mathbb{T}_{2}\). Let \(f (s_{1},t_{1}) \in CC_{\mathrm{rd}}(I_{x} \times I_{y} , {\mathbb{R}}^{+})\) and \(g (k_{1},r_{1}) \in CC_{\mathrm{rd}}(I_{z} \times I_{\omega }, {\mathbb{R}}^{+})\), and define

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

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

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{F (s_{1},t_{1})G (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant C_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl( \sigma (x)-s_{1}\bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[f (s_{1},t_{1}) \bigr]^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }\bigl( \sigma (z)-k_{1} \bigr) \bigl(\sigma (\omega ) -r_{1}\bigr) \bigl[g (k_{1},r_{1}) \bigr]^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$
(2.1)

where

$$ C_{1}(p_{1})= \bigl[(x-t_{0}) (y-t_{0}) (z-t_{0}) (\omega -t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}}. $$

Proof

By the assumptions, applying Hölder’s inequality with indices \(\frac{p_{1}}{p_{1}-1}\) and \(p_{1}\), we have

$$ F (s_{1},t_{1}) \leqslant \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[f (\xi ,\eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} $$
(2.2)

and

$$ G (k_{1},r_{1}) \leqslant \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[g (\xi ,\eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. $$
(2.3)

By multiplying (2.2) and (2.3) we get

$$\begin{aligned} F (s_{1},t_{1})G (k_{1},r_{1}) \leqslant& \bigl( \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] \bigr)^{\frac{p_{1}-1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[f (\xi , \eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[g (\xi , \eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.4)

Using Lemma 1.9 (for \(x,y\ge 0\)) in (2.4), we get

$$\begin{aligned} F (s_{1},t_{1})G (k_{1},r_{1}) \leqslant& \bigl(h \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr]+ h^{*} \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] \bigr)^{\frac{p_{1}-1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[f (\xi , \eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[g (\xi , \eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.5)

Using Lemma 1.4 in (2.5) gives

$$\begin{aligned} F (s_{1},t_{1})G (k_{1},r_{1}) \leqslant & \bigl( \bigl\vert h \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] \bigr\vert ^{\frac{1}{2 \beta }}+ \bigl\vert h^{*} \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] \bigr\vert ^{\frac{1}{2 \beta }} \bigr)^{\frac{2 \alpha (p_{1}-1)}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[f ( \xi ,\eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[g ( \xi ,\eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.6)

Dividing both sides of (2.6) by \(( |h [(s_{1}-t_{0})(t_{1}-t_{0}) ] |^{\frac{1}{2 \beta }}+ |h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] |^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}\), we get

$$\begin{aligned}& \frac{F (s_{1},t_{1})G (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \\& \quad \leqslant \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[f ( \xi ,\eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[g (\xi ,\eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.7)

Integrating both sides of (2.7) firstly with respect to \(r_{1}\) and \(k_{1}\) and then with respect to \(s_{1}\) and \(t_{1}\), and applying Hölder’s inequality with indices \(\frac{p_{1}}{p_{1}-1}\) and \(p_{1}\), we obtain

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{F (s_{1},t_{1})G (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant \bigl[(x-t_{0}) (y-t_{0}) (z-t_{0}) (\omega -t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[f (\xi ,\eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)\Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[g (\xi ,\eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr) \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}} \\& \quad = C_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl[f (\xi ,\eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr)\Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl[g (\xi ,\eta ) \bigr]^{p_{1}} \Delta \xi \Delta \eta \biggr) \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.8)

Applying Fubini’s theorem to the right-hand side of (2.8), we have

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{F (s_{1},t_{1})G (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant C_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} (x-s_{1}) (y-t_{1}) \bigl[f (s_{1},t_{1}) \bigr]^{p} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }(z-k_{1}) ( \omega -r_{1}) \bigl[g (k_{1},r_{1}) \bigr]^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$

Using the relations \(\sigma (x)\geqslant x\), \(\sigma (y)\geqslant y\), \(\sigma (\omega)\geq \omega\), and \(\sigma (z)=z\), we obtain

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{F (s_{1},t_{1})G (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant C_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl( \sigma (x)-s_{1}\bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[f (s_{1},r_{1}) \bigr]^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }\bigl( \sigma (z)-k_{1} \bigr) \bigl(\sigma (\omega ) -r_{1}\bigr) \bigl[g (k_{1},t_{1}) \bigr]^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$

This completes the proof. □

In the particular case of Theorem 2.1 where \(\mathbb{T}_{1}=\mathbb{T}_{1}=\mathbb{R}\), we have \(\sigma (y)=y\), \(\sigma (x)=x\), \(\sigma (\omega)=\omega\), and \(\sigma (z)=z\), and we get the following result.

Corollary 2.2

Let \(f (s_{1},t_{1})\) and \(g (k_{1},r_{1})\) be real-valued continuous functions, and define

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

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

$$\begin{aligned}& \int _{0}^{x} \int _{0}^{y} \biggl( \int _{0}^{z} \int _{0}^{\omega }\frac{F (s_{1},t_{1})G (k_{1},r_{1})}{ ( \vert h [(s_{1})(t_{1}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1})(r_{1}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \,dk_{1} \,dr_{1} \biggr)\,ds_{1} \,dt_{1} \\& \quad \leqslant C ^{*}_{1}(p_{1}) \biggl( \int _{0}^{x} \int _{0}^{y} (x-s_{1}) (y-t_{1}) \bigl[f (s_{1},t_{1}) \bigr]^{p_{1}} \,ds_{1} \,dt_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{0}^{z} \int _{0}^{\omega }(z-k_{1}) ( \omega -r_{1}) \bigl[g (k_{1},r_{1}) \bigr]^{p_{1}} \,dk_{1} \,dr_{1} \biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$

where

$$ C^{*} _{1}(p_{1})= \bigl[(x) (y) (z) (\omega ) \bigr]^{\frac{p_{1}-1}{p_{1}}}. $$

In the particular case of Theorem 2.1 where \(\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{Z}\), we have \(\sigma (x)=x+1\), \(\sigma (y)=y+1\), \(\sigma (w)=\omega+1\), and \(\sigma (z)=z+1\), and we get the following result.

Corollary 2.3

Let \(\{a_{m_{1},n_{1}}\}_{0 \leqslant {m_{1},n_{1}} \leqslant N}\) and \(\{b_{k_{1},r_{1}}\}_{0 \leqslant {k_{1},r_{1}} \leqslant N}\) be nonnegative sequences of real numbers, and define

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

Then

$$\begin{aligned}& \sum_{s_{1}=1}^{m_{1}} \sum _{t_{1}=1}^{n_{1}} \Biggl(\sum _{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{\omega _{1}} \frac{A_{s_{1},t_{1}}B_{k_{1},r_{1}}}{ ( \vert h(s_{1} t_{1}) \vert ^{\frac{1}{2\beta }}+ \vert h^{*}(k_{1} r_{1}) \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Biggr) \\& \quad \leqslant C _{2}(p_{1}) \Biggl(\sum _{s_{1}=1}^{m_{1}} \sum_{t_{1}=1}^{n_{1}} \bigl((m_{1}+1)-s_{1}\bigr) \bigl((n_{1}+1)-t_{1} \bigr) (a_{s_{1},t_{1}})^{p_{1}} \Biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \Biggl(\sum_{k_{1}=1}^{z_{1}} \sum _{r_{1}=1}^{ \omega _{1}}\bigl((z_{1}+1)-k_{1} \bigr) \bigl((\omega _{1}+1)-r_{1}\bigr) (b_{k_{1},r_{1}})^{p_{1}} \Biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$

where

$$ C _{2}(p_{1})=(m_{1} n_{1} z_{1} \omega _{1})^{\frac{p_{1}-1}{p_{1}}}. $$

Corollary 2.4

Under the assumptions of Theorem 2.1, we have

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{F (s_{1},t_{1})G (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant C_{1}(p_{1}) \biggl\{ h \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl[f (s_{1} , t_{1} ) \bigr]^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr) \\& \qquad {}+ h^{*} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }\bigl( \sigma (z)-k_{1} \bigr) \bigl(\sigma (\omega ) -r_{1}\bigr) \bigl[g (k_{1} ,r_{1} ) \bigr]^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr) \biggr\} ^{\frac{1}{p_{1}}}. \end{aligned}$$

Proof

Using (1.7) in (2.1), we get the desired result. □

Theorem 2.5

Under the assumptions of Theorem 2.1, let \(p (\xi ,\eta )\) and \(q (\xi ,\eta )\) be two positive functions. Let \(\Psi \geq 0\) and \(\Phi \geq 0\) be submultiplicative convex functions on \([0,\infty )\). Define

$$ P (s_{1},t_{1}):= \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} p (\xi , \eta ) \Delta \xi \Delta \eta \quad \textit{and}\quad Q (k_{1},r_{1}):= \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} q (\xi ,\eta ) \Delta \xi \Delta \eta . $$

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

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi (F (s_{1},t_{1}))\Psi (G (k_{1},r_{1}))}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant D_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl( \sigma (x)-s_{1}\bigr) \bigl(\sigma (y)-t_{1}\bigr) \biggl(p (s_{1},t_{1})\Phi \biggl[ \frac{f (s_{1},t_{1})}{p (s_{1},t_{1})} \biggr] \biggr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggr( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \bigl( \sigma (z)-k_{1}\bigr) \bigl(\sigma (\omega )-r_{1}\bigr) \biggl(q (k_{1},r_{1})\Psi \biggl[\frac{g (k_{1},r_{1})}{q (k_{1} ,r_{1} )} \biggr] \biggr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$
(2.9)

where

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

Proof

Since Φ is a convex submultiplicative function, by applying Jensen’s inequality we get that

$$\begin{aligned} \Phi \bigl(F (s_{1},t_{1}) \bigr) =& \Phi \biggl( \frac{P (s_{1},t_{1}) \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} p (\xi ,\eta )\frac{f (\xi ,\eta )}{p (\xi ,\eta )}\Delta \xi \Delta \eta }{ \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} p (\xi ,\eta )\Delta \xi \Delta \eta } \biggr) \\ \leqslant & \Phi \bigl(P (s_{1},t_{1}) \bigr)\Phi \biggl( \frac{ \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} p (\xi ,\eta )\frac{f (\xi ,\eta )}{p (\xi ,\eta )} \Delta \xi \Delta \eta }{ \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} p (\xi ,\eta )\Delta \xi \Delta \eta } \biggr) \\ \leqslant & \frac{ \Phi (P (s_{1},t_{1}))}{P (s_{1},t_{1})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} p (\xi ,\eta )\Phi \biggl( \frac{f (\xi ,\eta )}{p (\xi ,\eta )} \biggr) \Delta \xi \Delta \eta . \end{aligned}$$
(2.10)

From Hölder’s inequality with indices \(\frac{p_{1}}{p_{1}-1}\) and \(p_{1}\) we have

$$\begin{aligned} \Phi \bigl(F (s_{1},t_{1})\bigr) \leqslant& \frac{ \Phi (P (s_{1},t_{1}))}{P (s_{1},t_{1})} \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl(p (\xi ,\eta )\Phi \biggl[ \frac{f (\xi ,\eta )}{p (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.11)

Analogously,

$$\begin{aligned} \Psi \bigl(G (k_{1},r_{1})\bigr) \leqslant& \frac{ \Psi (Q (k_{1},r_{1}))}{Q (k_{1},r_{1})} \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}} \\ &{}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl(q (\xi ,\eta )\Psi \biggl[ \frac{g (\xi ,\eta )}{q (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.12)

From (2.11) and (2.12) we have

$$\begin{aligned}& \Phi \bigl(F (s_{1},t_{1})\bigr)\Psi \bigl(G (s_{1},t_{1})\bigr) \\& \quad \leqslant \bigl( \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] \bigr)^{\frac{p_{1}-1}{p_{1}}} \\& \qquad {} \times \biggl(\frac{ \Phi (P (s_{1},t_{1}))}{P (s_{1},t_{1})} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl(p (\xi , \eta )\Phi \biggl[\frac{f (\xi ,\eta )}{p (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggr) \\& \qquad {} \times \biggl(\frac{ \Psi (Q (k_{1},r_{1}))}{Q (k_{1},r_{1})} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl(q (\xi , \eta )\Psi \biggl[\frac{g (\xi ,\eta )}{q (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggr). \end{aligned}$$
(2.13)

Applying (1.7) to the term \(( [(s_{1}-t_{0})(t_{1}-t_{0}) ] [(k_{1}-t_{0})(r_{1}-t_{0}) ] )^{\frac{p_{1}-1}{p_{1}}}\), we get the inequality

$$\begin{aligned}& \Phi \bigl(F (s_{1},t_{1})\bigr)\Psi \bigl(G (k_{1},r_{1})\bigr) \\& \quad \leqslant \bigl(h \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr]+h^{*} \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] \bigr)^{\frac{p_{1}-1}{p_{1}}} \\& \qquad {} \times \biggl(\frac{ \Phi (P (s_{1},t_{1}))}{P (s_{1},t_{1})} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl(p (\xi , \eta )\Phi \biggl[\frac{f (\xi ,\eta )}{p (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggr) \\& \qquad {} \times \biggl(\frac{ \Psi (Q (k_{1},r_{1}))}{Q (k_{1},r_{1})} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl(q (\xi , \eta )\Psi \biggl[\frac{g (\xi ,\eta )}{q (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggr). \end{aligned}$$
(2.14)

Applying Lemma 1.4, we have

$$\begin{aligned}& \Phi \bigl(F (s_{1},t_{1})\bigr)\Psi \bigl(G (k_{1},r_{1})\bigr) \\& \quad \leqslant \bigl( \bigl\vert h \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] \bigr\vert ^{\frac{1}{2 \beta }}+ \bigl\vert h^{*} \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] \bigr\vert ^{\frac{1}{2 \beta }} \bigr)^{\frac{2 \alpha (p_{1}-1)}{p_{1}}} \\& \qquad {} \times \biggl(\frac{ \Phi (P (s_{1},t_{1}))}{P (s_{1},t_{1})} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl(p (\xi , \eta )\Phi \biggl[\frac{f (\xi ,\eta )}{p (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggr) \\& \qquad {} \times \biggl(\frac{ \Psi (Q (k_{1},r_{1}))}{Q (k_{1},r_{1})} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl(q (\xi , \eta )\Psi \biggl[\frac{g (\xi ,\eta )}{q (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggr). \end{aligned}$$
(2.15)

From (2.15) we have

$$\begin{aligned}& \frac{\Phi (F (s_{1},t_{1}))\Psi (G (k_{1},r_{1}))}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \\& \quad \leqslant \biggl(\frac{ \Phi (P (s_{1},t_{1}))}{P (s_{1},t_{1})} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl(p (\xi , \eta )\Phi \biggl[\frac{f (\xi ,\eta )}{p (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggr) \\& \qquad {} \times \biggl( \frac{ \Psi (Q (k_{1},r_{1}))}{Q (k_{1},r_{1})} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl(q (\xi ,\eta )\Psi \biggl[ \frac{g (\xi ,\eta )}{q (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggr). \end{aligned}$$
(2.16)

Integrating both sides of (2.16) firstly with respect to \(r_{1}\) and \(k_{1}\) and then with respect to \(s_{1}\) and \(t_{1}\), we get

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi (F (s_{1},t_{1}))\Psi (G (k_{1},r_{1}))}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \frac{ \Phi (P (s_{1},t_{1}))}{P (s_{1},t_{1})} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl(p (\xi ,\eta )\Phi \biggl[ \frac{f (\xi ,\eta )}{p (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \Delta s_{1} \Delta t_{1} \biggr) \\& \qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }\frac{ \Psi (Q (k_{1},r_{1}))}{Q (k_{1},r_{1})} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl(q (\xi ,\eta )\Psi \biggl[ \frac{g (\xi ,\eta )}{q (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}\Delta k_{1} \Delta r_{1} \biggr). \end{aligned}$$
(2.17)

From Hölder’s inequality with indices \(p_{1}\) and \(\frac{p_{1}}{p_{1}-1}\) we have

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi (F (s_{1},t_{1}))\Psi (G (k_{1},r_{1}))}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \frac{ \Phi (P (s_{1},t_{1}))}{P (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}}^{\omega }\biggl(\frac{ \Psi (Q (k_{1},r_{1}))}{Q (k_{1},r_{1})} \biggr)^{\frac{p_{1}}{p_{1}-1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{p_{1}-1}{p_{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 (\xi ,\eta )\Phi \biggl[ \frac{f (\xi ,\eta )}{p (\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}}^{\omega } \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl(q (\xi ,\eta )\Psi \biggl[ \frac{g (\xi ,\eta )}{q (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)\Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}} \\& \quad =D_{1}(p) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \biggl(p (\xi ,\eta )\Phi \biggl[ \frac{f (\xi ,\eta )}{p (\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}}^{\omega } \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \biggl(q (\xi ,\eta )\Psi \biggl[ \frac{g (\xi ,\eta )}{q (\xi ,\eta )} \biggr] \biggr)^{p_{1}} \Delta \xi \Delta \eta \biggr)\Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.18)

Applying Fubini’s theorem to (2.18), we obtain

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi (F (s_{1},t_{1}))\Psi (G (k_{1},r_{1}))}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p-1)}{p}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant D_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} (x-s_{1}) (y-t_{1}) \biggl(p (s_{1},t_{1})\Phi \biggl[ \frac{f (s_{1},t_{1})}{p (s_{1},t_{1})} \biggr] \biggr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggr( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } (z-k_{1}) ( \omega -r_{1}) \biggl(q (k_{1},r_{1})\Psi \biggl[ \frac{g (k_{1},r_{1})}{q (k_{1} ,r_{1} )} \biggr] \biggr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$

From the relations \(\sigma (x)\geqslant x\), \(\sigma (y)\geqslant y\), \(\sigma (\omega)\geq \omega\), and \(\sigma (z)=z\) we obtain

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi (F (s_{1},t_{1}))\Psi (G (k_{1},r_{1}))}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant D_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl( \sigma (x)-s_{1}\bigr) \bigl(\sigma (y)-t_{1}\bigr) \biggl(p (s_{1},t_{1})\Phi \biggl[ \frac{f (s_{1},t_{1})}{p (s_{1},t_{1})} \biggr] \biggr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggr( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \bigl( \sigma (z)-k_{1}\bigr) \bigl(\sigma (\omega )-r_{1}\bigr) \biggl(q (k_{1},r_{1})\Psi \biggl[\frac{g (k_{1},r_{1})}{q (k_{1} ,r_{1} )} \biggr] \biggr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$

where

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

This completes the proof. □

Taking \(\mathbb{T}_{1}=\mathbb{T}_{1}=\mathbb{R}\) in Theorem 2.5, we have \(\sigma (x)=x\), \(\sigma (y)=y\), \(\sigma (\omega)=\omega\), \(\sigma (z)=z\), and we get the following result.

Corollary 2.6

Let \(f (s_{1},t_{1})\) and \(g (k_{1},r_{1})\) be real-valued continuous functions, and let \(p (s_{1},t_{1})\) and \(q (k_{1},r_{1})\) be positive functions. Define

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

Then

$$\begin{aligned}& \int _{0}^{x} \int _{0}^{y} \biggl( \int _{0}^{z} \int _{0}^{\omega }\frac{\Phi (F (s_{1},t_{1}))\Psi (G (k_{1},r_{1}))}{ ( \vert h [s_{1} t_{1} ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [k_{1} r_{1} ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \,dk_{1} \,dr_{1} \biggr)\,ds_{1} \,dt_{1} \\& \quad \leqslant D ^{*}_{1}(p_{1}) \biggl( \int _{0}^{x} \int _{0}^{y} (x-s_{1}) (y-t_{1}) \biggl(p (s_{1},t_{1})\Phi \biggl[ \frac{f (s_{1},t_{1})}{p (s_{1},t_{1})} \biggr] \biggr)^{p_{1}} \,ds_{1} \,dt_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {}\times \biggr( \int _{0}^{z} \int _{0}^{\omega } (z-k_{1}) ( \omega -r_{1}) \biggl(q (k_{1},r_{1})\Psi \biggl[ \frac{g (k_{1},r_{1})}{q (k_{1} ,r_{1} )} \biggr] \biggr)^{p_{1}} \,dk_{1} \,dr_{1} \biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$

where

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

In the particular case of Theorem 2.5 where \(\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{Z}\), we have \(\sigma (x)=x+1\), \(\sigma (y)=y+1\), \(\sigma (\omega)=\omega+1\), \(\sigma (z)=z+1\), and we get the following result.

Corollary 2.7

Let \(\{a_{m_{1},n_{1}}\}_{0 \leqslant m_{1},n_{1} \leqslant N}\) and \(\{b_{k_{1},r_{1}}\}_{0 \leqslant k_{1},r_{1} \leqslant N}\) be nonnegative sequences of real numbers, and let \(\{p_{m_{1},n_{1}}\}_{0 \leqslant m_{1},n_{1} \leqslant N}\) be \(\{q_{k_{1},r_{1}}\}_{0 \leqslant k_{1},r_{1} \leqslant N}\) positive sequences. Define

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

Then

$$\begin{aligned}& \sum_{s_{1}=1}^{m_{1}} \sum _{t_{1}=1}^{n_{1}} \Biggl(\sum _{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{\omega _{1}} \frac{ \Phi (A_{s_{1},t_{1}}) \Psi (B_{k_{1},r_{1}})}{ ( \vert h(s_{1} t_{1}) \vert ^{\frac{1}{2\beta }}+ \vert h^{*}(k_{1} r_{1}) \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Biggr) \\& \quad \leqslant D^{**}(p_{1}) \Biggl\{ \sum _{s_{1}=1}^{m_{1}} \sum_{t_{1}=1}^{n_{1}} \bigl((m_{1}+1)-s_{1}\bigr) \bigl((n_{1}+1)-t_{1} \bigr) \biggl(p_{s_{1},t_{1}} \Phi \biggl[ \frac{a_{s_{1},t_{1}}}{p_{s_{1},t_{1}}} \biggr] \biggr)^{p_{1}} \Biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {} \times \Biggl\{ \sum_{k_{1}=1}^{z_{1}} \sum _{r_{1}=1}^{\omega _{1}}\bigl((z_{1}+1)-k_{1} \bigr) \bigl(( \omega _{1}+1)-r_{1}\bigr) \biggl(q_{k_{1},r_{1}} \Psi \biggl[ \frac{b_{k_{1},r_{1}}}{q_{k_{1},r_{1}}} \biggr] \biggr)^{p_{1}} \Biggr\} ^{\frac{1}{p_{1}}}, \end{aligned}$$

where

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

Corollary 2.8

Under the assumptions of Theorem 2.5, we have

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi (F (s_{1},t_{1}))\Psi (G (k_{1},r_{1}))}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant D_{1}(p_{1}) \biggl\{ h \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \biggl(p (s_{1},t_{1}) \Phi \biggl[ \frac{f (s_{1},t_{1})}{p (s_{1},t_{1})} \biggr] \biggr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr) \\& \qquad {}+h^{*} \biggr( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \bigl( \sigma (z)-k_{1}\bigr) \bigl(\sigma (\omega )-r_{1}\bigr) \biggl(q (k_{1},r_{1})\Psi \biggl[\frac{g (k_{1},r_{1})}{q (k_{1} ,r_{1} )} \biggr] \biggr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr) \biggr\} ^{\frac{1}{p_{1}}}. \end{aligned}$$

Proof

Using (1.7) in (2.9), we get the desired result. □

Theorem 2.9

Under the assumptions of Theorem 2.5, define

$$ \begin{aligned} &F (s_{1},t_{1}):= \frac{1}{(s_{1}-t_{0})(t_{1}-t_{0})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} f (\xi ,\eta ) \Delta \xi \Delta \eta, \\ &G (k_{1},r_{1}):= \frac{1}{(k_{1}-t_{0})(r_{1}-t_{0})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} g (\xi ,\eta ) \Delta \xi \Delta \eta . \end{aligned} $$
(2.19)

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

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))(s_{1}-t_{0})(t_{1}-t_{0}) (k_{1}-t_{0})(r_{1}-t_{0})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant K_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl(\Phi \bigl(f (s_{1} ,t_{1} ) \bigr) \bigr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (\omega )-r_{1}\bigr) \bigl(\Psi \bigl(g (k_{1} , r_{1} )\bigr) \bigr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$
(2.20)

where

$$ K_{1}(p_{1})= \bigl[(x-t_{0}) (y-t_{0}) (z-t_{0}) (\omega -t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}}. $$

Proof

From (2.19), using Jensen’s inequality, we see that

$$\begin{aligned} \Phi \bigl( F (s_{1},t_{1}) \bigr) = &\Phi \biggl( \frac{1}{(s_{1}-t_{0})(t_{1}-t_{0})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} f (\xi ,\eta ) \Delta \xi \Delta \eta \biggr) \\ \leqslant &\frac{1}{(s_{1}-t_{0})(t_{1}-t_{0})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \Phi \bigl(f (\xi ,\eta )\bigr) \Delta \xi \Delta \eta . \end{aligned}$$
(2.21)

Similarly,

$$\begin{aligned} \Psi \bigl( G (k_{1},r_{1}) \bigr) = &\Psi \biggl( \frac{1}{(k_{1}-t_{0})(r_{1}-t_{0})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} g (\xi ,\eta ) \Delta \xi \Delta \eta \biggr) \\ \leqslant& \frac{1}{(k_{1}-t_{0})(r_{1}-t_{0})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \Psi \bigl(g (\xi ,\eta )\bigr) \Delta \xi \Delta \eta . \end{aligned}$$
(2.22)

By multiplying (2.21) and (2.22) we get

$$\begin{aligned}& \Phi \bigl( F (s_{1},t_{1}) \bigr)\Psi \bigl( G (k_{1},r_{1})\bigr) \\& \quad \leqslant \frac{1}{(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0})} \\& \qquad {} \times \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \Phi \bigl(f (\xi , \eta )\bigr) \Delta \xi \Delta \eta \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \Psi \bigl(g (\xi ,\eta )\bigr) \Delta \xi \Delta \eta . \end{aligned}$$
(2.23)

This implies that

$$\begin{aligned}& \Phi \bigl( F (s_{1},t_{1}) \bigr)\Psi \bigl( G (k_{1},r_{1})\bigr) (s_{1}-t_{0}) (t_{1}-t_{0}) (k_{1}-t_{0}) (r_{1}-t_{0}) \\& \quad \leqslant \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \Phi \bigl(f (\xi , \eta )\bigr) \Delta \xi \Delta \eta \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \Psi \bigl(g (\xi ,\eta )\bigr) \Delta \xi \Delta \eta . \end{aligned}$$
(2.24)

Using Hölder’s inequality with indices \(p_{1}\) and \(\frac{p_{1}}{p_{1}-1}\), we have

$$\begin{aligned}& \Phi \bigl( F (s_{1},t_{1}) \bigr)\Psi \bigl( G (k_{1},r_{1})\bigr) (s_{1}-t_{0}) (t_{1}-t_{0}) (k_{1}-t_{0}) (r_{1}-t_{0}) \\& \quad \leqslant \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) (k_{1}-t_{0}) (r_{1}-t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(\Phi \bigl(f (\xi ,\eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(\Psi \bigl(g (\xi ,\eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.25)

Applying Lemma 1.9 to the term \([(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0}) ]^{\frac{p_{1}-1}{p_{1}}}\), we get

$$\begin{aligned}& \Phi \bigl( F (s_{1},t_{1}) \bigr)\Psi \bigl( G (k_{1},r_{1})\bigr) (s_{1}-t_{0}) (t_{1}-t_{0}) (k_{1}-t_{0}) (r_{1}-t_{0}) \\& \quad \leqslant \bigl[h \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr]+h^{*} \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] \bigr]^{\frac{p_{1}-1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(\Phi \bigl(f (\xi ,\eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(\Psi \bigl(g (\xi ,\eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.26)

Applying Lemma 1.4 to (2.26), we obtain

$$\begin{aligned}& \Phi \bigl( F (s_{1},t_{1}) \bigr)\Psi \bigl( G (k_{1},r_{1})\bigr) (s_{1}-t_{0}) (t_{1}-t_{0}) (k_{1}-t_{0}) (r_{1}-t_{0}) \\& \quad \leqslant \bigl[ \bigl\vert h \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] \bigr\vert ^{\frac{1}{2\beta }}+ \bigl\vert h^{*} \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] \bigr\vert ^{\frac{1}{2\beta }} \bigr]^{\frac{2 \alpha (p_{1}-1)}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(\Phi \bigl(f (\xi ,\eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(\Psi \bigl(g (\xi ,\eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.27)

Dividing both sides of (2.27) by \([|h [(s_{1}-t_{0})(t_{1}-t_{0}) ]|^{\frac{1}{2\beta }}+|h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ]|^{\frac{1}{2\beta }} ]^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}\), we get

$$\begin{aligned}& \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))(s_{1}-t_{0})(t_{1}-t_{0}) (k_{1}-t_{0})(r_{1}-t_{0})}{ [ \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2\beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2\beta }} ]^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \\& \quad \leqslant \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl( \Phi \bigl(f (\xi ,\eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(\Psi \bigl(g (\xi ,\eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.28)

Integrating both sides of (2.28) firstly with respect to \(r_{1}\) and \(k_{1}\) and then with respect to \(s_{1}\) and \(t_{1}\) and using Hölder’s inequality with indices \(\frac{p_{1}}{p_{1}-1}\) and \(p_{1}\), we get

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant \bigl[(x-t_{0}) (y-t_{0}) (z-t_{0}) (\omega -t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(\Phi \bigl(f (\xi , \eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)\Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(\Psi \bigl(g (\xi ,\eta ) \bigr) \bigr)^{p}_{1} \Delta \xi \Delta \eta \biggr) \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}} \\& \quad =K_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(\Phi \bigl(f (\xi ,\eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)\Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(\Psi \bigl(g (\xi ,\eta ) \bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr) \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.29)

Applying Fubini’s theorem to (2.29), we have

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))(s_{1}-t_{0})(t_{1}-t_{0}) (k_{1}-t_{0})(r_{1}-t_{0})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant K_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} (x-s_{1}) (y-t_{1}) \bigl(\Phi \bigl(f (s_{1} ,t_{1} )\bigr) \bigr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }(z-k_{1}) ( \omega -r_{1}) \bigl(\Psi \bigl(g (k_{1} ,r_{1} )\bigr) \bigr)^{p_{1} } \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$

From the relations \(\sigma (x)\geqslant x\), \(\sigma (\omega)\geq \omega\), and \(\sigma (z)\geq z\) we obtain

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant K_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl(\Phi \bigl(f (s_{1} ,t_{1} ) \bigr) \bigr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (\omega )-r_{1}\bigr) \bigl(\Psi \bigl(g (k_{1} , r_{1} )\bigr) \bigr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$

where

$$ K_{1}(p_{1})= \bigl[(x-t_{0}) (y-t_{0}) (z-t_{0}) (\omega -t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}}. $$

This completes the proof. □

Taking \(\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{R}\) and \(\sigma (x)=x\), \(\sigma (y)=y\), \(\sigma (\omega)=\omega\), and \(\sigma (z)=z\), by Theorem 2.9 we obtain the following corollaries.

Corollary 2.10

Let \(f (s_{1},t_{1})\) and \(g (k_{1},r_{1})\) be real-valued continuous functions, and define

$$ F (s_{1},t_{1}):=\frac{1}{s_{1} t_{1}} \int _{0}^{s_{1}} \int _{0}^{t_{1}} f (\xi ,\tau ) \,d\xi \,d\tau \quad \textit{and}\quad G (k_{1},r_{1}):= \frac{1}{k_{1} r_{1}} \int _{0}^{k_{1}} \int _{0}^{r_{1}} g (\xi , \tau ) \,d\xi \,d\tau . $$

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

$$\begin{aligned}& \int _{0}^{x} \int _{0}^{y} \biggl( \int _{0}^{z} \int _{0}^{\omega }\frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))(s_{1}t_{1})(k_{1} r_{1})}{ ( \vert h [(s_{1}t_{1}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}r_{1}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \,dk_{1} \,dr_{1} \biggr)\,ds_{1} \,dt_{1} \\& \quad \leqslant K^{*} _{1}(p_{1}) \biggl( \int _{0}^{x} \int _{0}^{y} (x-s_{1}) (y-t_{1}) \bigl( \Phi \bigl(f (s_{1} ,t_{1} )\bigr) \bigr)^{p_{1}} \,ds_{1} \,dt_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{0}^{z} \int _{0}^{\omega }(z-k_{1}) ( \omega -r_{1}) \bigl(\Psi \bigl(g (k_{1} ,r_{1} )\bigr) \bigr)^{p_{1}} \,dk_{1} \,dr_{1} \biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$

where

$$ K^{*} _{1}(p_{1})= \bigl[(x) (y) (z) (\omega ) \bigr]^{\frac{p_{1}-1}{p_{1}}}. $$

Taking \(\mathbb{T}_{1}=\mathbb{T}_{2}= \mathbb{Z}\) in Theorem 2.9, we have \(\sigma (x)=x+1\), \(\sigma (y)=y+1\), \(\sigma (\omega)=\omega+1\), and \(\sigma (z)=z+1\), and we get the following result.

Corollary 2.11

Let \(\{a_{m_{1},n_{1}}\}_{0 \leqslant m_{1},n_{1} \leqslant N}\) and \(\{b_{k_{1},r_{1}}\}_{0 \leqslant k_{1},r_{1} \leqslant N}\) be nonnegative sequences of real numbers, and define

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

Then

$$\begin{aligned}& \sum_{s_{1}=1}^{m_{1}} \sum_{t_{1}=1}^{n_{1}} \Biggl(\sum _{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{\omega _{1}} \frac{(s_{1} t_{1})(k_{1} r_{1})\Phi ( A_{s_{1},t_{1}}) \Psi (B_{k_{1},r_{1}})}{ ( \vert h(s_{1} t_{1}) \vert ^{\frac{1}{2\beta }}+ \vert h^{*}(k_{1} r_{1}) \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Biggr) \\& \quad \leqslant K^{**}(p_{1}) \Biggl\{ \sum _{s_{1}=1}^{m_{1}} \sum_{t_{1}=1}^{n_{1}} \bigl((m_{1}+1)-s_{1}\bigr) \bigl((n_{1}+1)-t_{1} \bigr) \bigl( \Phi ( a_{s_{1},t_{1}}) \bigr)^{p_{1}} \Biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {} \times \Biggl\{ \sum_{k_{1}=1}^{z_{1}} \sum _{r_{1}=1}^{\omega _{1}}\bigl((z_{1}+1)-k_{1} \bigr) \bigl(( \omega _{1}+1)-r_{1}\bigr) \bigl( \Psi ( b_{k_{1},r_{1}}) \bigr)^{p_{1}} \Biggr\} ^{\frac{1}{p_{1}}}, \end{aligned}$$

where

$$ K^{**}(p_{1})=(m_{1} n_{1} z_{1} \omega _{1})^{\frac{p_{1}-1}{p_{1}}}. $$

Corollary 2.12

Under the assumptions of Theorem 2.9, we have

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant K_{1}(p_{1}) \biggl\{ h \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl(\Phi \bigl(f (s_{1} ,t_{1} ) \bigr) \bigr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr) \\& \qquad {} +h^{*} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (\omega )-r_{1}\bigr) \bigl(\Psi \bigl(g (k_{1} , r_{1} )\bigr) \bigr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr) \biggr\} ^{\frac{1}{p_{1}}}. \end{aligned}$$

Proof

Using (1.7) in (2.20), we get the desired result. □

Theorem 2.13

Under the assumptions of Theorem 2.5, assume that

$$ \begin{aligned} &F (s_{1},t_{1}):= \frac{1}{P (s_{1},t_{1})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} p (\xi ,\eta ) f (\xi ,\eta ) \Delta \xi \Delta \eta, \\ &G (k_{1},r_{1}):= \frac{1}{Q (k_{1},r_{1})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} q (\xi ,\eta ) g (\xi ,\eta ) \Delta \xi \Delta \eta . \end{aligned} $$
(2.30)

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

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))P (s_{1},t_{1})Q (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant H_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl(p(s_{1},t_{1})\Phi \bigl(f (s_{1} ,t_{1} ) \bigr) \bigr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (\omega )-r_{1}\bigr) \bigl(q(k_{1},r_{1})\Psi \bigl(g (k_{1} , r_{1} )\bigr) \bigr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$
(2.31)

where

$$ H_{1}(p_{1})= \bigl[(x-t_{0}) (y-t_{0}) (z-t_{0}) (\omega -t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}}. $$

Proof

From (2.30), using Jensen’s inequality, we see that

$$\begin{aligned} \Phi \bigl( F (s_{1},t_{1}) \bigr) = &\Phi \biggl(\frac{1}{P (s_{1},t_{1})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}}p (\xi ,\eta ) f (\xi ,\eta ) \Delta \xi \Delta \eta \biggr) \\ \leqslant& \frac{1}{P (s_{1},t_{1})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}}p (\xi ,\eta ) \Phi \bigl(f (\xi , \eta )\bigr) \Delta \xi \Delta \eta . \end{aligned}$$
(2.32)

Similarly,

$$\begin{aligned} \Psi \bigl( G (k_{1},r_{1}) \bigr) = &\Psi \biggl(\frac{1}{Q (k_{1},r_{1})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} q (\xi ,\eta ) g (\xi ,\eta ) \Delta \xi \Delta \eta \biggr) \\ \leqslant& \frac{1}{Q (k_{1},r_{1})} \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}q (\xi ,\eta ) \Psi \bigl(g (\xi , \eta )\bigr) \Delta \xi \Delta \eta . \end{aligned}$$
(2.33)

By multiplying (2.32) and (2.33) we get

$$\begin{aligned} \Phi \bigl( F (s_{1},t_{1}) \bigr)\Psi \bigl( G (k_{1},r_{1})\bigr) \leqslant & \frac{1}{P (s_{1},t_{1})Q (k_{1},r_{1})} \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} p (\xi ,\eta )\Phi \bigl(f ( \xi ,\eta )\bigr) \Delta \xi \Delta \eta \\ &{} \times \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}q (\xi ,\eta ) \Psi \bigl(g (\xi , \eta )\bigr) \Delta \xi \Delta \eta . \end{aligned}$$
(2.34)

This implies that

$$\begin{aligned}& \Phi \bigl( F (s_{1},t_{1})\bigr) \Psi \bigl( G (k_{1},r_{1})\bigr) P (s_{1},t_{1})Q (k_{1},r_{1}) \\& \quad \leqslant \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} p (\xi ,\eta ) \Phi \bigl(f (\xi ,\eta )\bigr) \Delta \xi \Delta \eta \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}}q (\xi ,\eta ) \Psi \bigl(g (\xi , \eta )\bigr) \Delta \xi \Delta \eta . \end{aligned}$$
(2.35)

Using Hölder’s inequality with indices \(p_{1}\) and \(\frac{p_{1}}{p_{1}-1}\), we obtain

$$\begin{aligned}& \Phi \bigl( F (s_{1},t_{1}) \bigr)\Psi \bigl( G (k_{1},r_{1})\bigr) P (s_{1},t_{1})Q (k_{1},r_{1}) \\& \quad \leqslant\bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) (k_{1}-t_{0}) (r_{1}-t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(p (\xi ,\eta )\Phi \bigl(f (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(q (\xi ,\eta )\Psi \bigl(g (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.36)

Applying Lemma 1.9 to the term \([(s_{1}-t_{0})(t_{1}-t_{0})(k_{1}-t_{0})(r_{1}-t_{0}) ]^{\frac{p_{1}-1}{p_{1}}}\), we get

$$\begin{aligned}& \Phi \bigl( F (s_{1},t_{1}) \bigr)\Psi \bigl( G (k_{1},r_{1})\bigr) P (s_{1},t_{1})Q (k_{1},r_{1}) \\& \quad \leqslant \bigl[h \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr]+h^{*} \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] \bigr]^{\frac{p_{1}-1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(p (\xi ,\eta )\Phi \bigl(f (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(q (\xi ,\eta )\Psi \bigl(g (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.37)

Applying Lemma 1.4 to (2.37), we obtain

$$\begin{aligned}& \Phi \bigl( F (s_{1},t_{1}) \bigr)\Psi \bigl( G (k_{1},r_{1})\bigr)P (s_{1},t_{1})Q (k_{1},r_{1}) \\& \quad \leqslant \bigl[ \bigl\vert h \bigl[(s_{1}-t_{0}) (t_{1}-t_{0}) \bigr] \bigr\vert ^{\frac{1}{2\beta }}+ \bigl\vert h^{*} \bigl[(k_{1}-t_{0}) (r_{1}-t_{0}) \bigr] \bigr\vert ^{\frac{1}{2\beta }} \bigr]^{\frac{2 \alpha (p_{1}-1)}{p_{1}}} \\& \qquad {}\times \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(p ( \xi ,\eta )\Phi \bigl(f (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(q (\xi ,\eta )\Psi \bigl(g (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.38)

Dividing both sides of (2.27) by \([|h [(s_{1}-t_{0})(t_{1}-t_{0}) ]|^{\frac{1}{2\beta }}+|h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ]|^{\frac{1}{2\beta }} ]^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}\), we get

$$\begin{aligned}& \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))P (s_{1},t_{1})Q (k_{1},r_{1})}{ [ \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2\beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2\beta }} ]^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \\& \quad \leqslant \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(p ( \xi ,\eta )\Phi \bigl(f (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(q (\xi ,\eta )\Psi \bigl(g (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)^{\frac{1}{p_{1}}} . \end{aligned}$$
(2.39)

Integrating both sides of (2.39) firstly with respect to \(r_{1}\) and \(k_{1}\) and then with respect to \(s_{1}\) and \(t_{1}\) and using Hölder’s inequality with indices \(\frac{p_{1}}{p_{1}-1}\) and \(p_{1}\), we get

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))P (s_{1},t_{1})Q (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant \bigl[(x-t_{0}) (y-t_{0}) (z-t_{0}) (\omega -t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(p (\xi ,\eta ) \Phi \bigl(f (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr) \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(q (\xi ,\eta )\Psi \bigl(g (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr) \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}} \\& \quad =H_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{s_{1}} \int _{t_{0}}^{t_{1}} \bigl(p (\xi ,\eta )\Phi \bigl(f (\xi ,\eta )\bigr) \bigr)^{p_{1}} \Delta \xi \Delta \eta \biggr)\Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \biggl( \int _{t_{0}}^{k_{1}} \int _{t_{0}}^{r_{1}} \bigl(q (\xi ,\eta )\Psi \bigl(g (\xi ,\eta )\bigr) \bigr)^{p_{1} } \Delta \xi \Delta \eta \biggr) \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$
(2.40)

Applying Fubini’s theorem, we get

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))P (s_{1},t_{1})Q (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant H_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} (x-s_{1}) (y-t_{1}) \bigl(p (s_{1} ,t_{1})\Phi \bigl(f (s_{1} ,t_{1} )\bigr) \bigr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }(z-k_{1}) ( \omega -r_{1}) \bigl(q (k_{1} ,r_{1} )\Psi \bigl(g (k_{1} ,r_{1} )\bigr) \bigr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}. \end{aligned}$$

From the relations \(\sigma (x)\geqslant x\), \(\sigma (y)\geqslant y\), \(\sigma (\omega)\geq \omega\), and \(\sigma (z)\geq z\) we obtain

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))P (s_{1},t_{1})Q (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant H_{1}(p_{1}) \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl(p (s_{1} ,t_{1} )\Phi \bigl(f ( s_{1} ,t_{1} )\bigr) \bigr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (\omega )-r_{1}\bigr) \bigl(q (k_{1} ,r_{1} )\Psi \bigl(g (k_{1} ,r_{1} )\bigr) \bigr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr)^{\frac{1}{p_{1}}}, \end{aligned}$$

where

$$ H_{1}(p_{1})= \bigl[(x-t_{0}) (y-t_{0}) (z-t_{0}) (\omega -t_{0}) \bigr]^{\frac{p_{1}-1}{p_{1}}}. $$

This completes the proof. □

Taking \(\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{R}\) in Theorem 2.9, we have \(\sigma (x)=x\), \(\sigma (y)=y\), \(\sigma (\omega)=\omega\), \(\sigma (z)=z\), and we get the following result.

Corollary 2.14

Let \(f (s_{1},t_{1})\), \(g (k_{1},r_{1})\) be real-valued continuous functions, and let \(p (s_{1},t_{1})\), \(q (r_{1},k_{1})\) be positive functions. Define

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

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

$$\begin{aligned}& \int _{0}^{x} \int _{0}^{y} \biggl( \int _{0}^{z} \int _{0}^{\omega }\frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))P (s_{1},t_{1})Q (k_{1},r_{1})}{ ( \vert h [(s_{1} t_{1}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}r_{1}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \,dk_{1} \,dr_{1} \biggr)\,ds_{1} \,dt_{1} \\& \quad \leqslant H^{*}_{1}(p_{1}) \biggl( \int _{0}^{x} \int _{0}^{y} (x-s_{1}) (y-t_{1}) \bigl(p (s_{1},t_{1}) \Phi \bigl(f (s_{1} ,t_{1} )\bigr) \bigr)^{p_{1}} \,ds_{1} \,dt_{1} \biggr)^{\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{0}^{z} \int _{0}^{\omega }(z-k_{1}) ( \omega -r_{1}) \bigl( q (k_{1},r_{1})\Psi \bigl(g (k_{1} ,r_{1})\bigr) \bigr)^{p_{1}} \,dk_{1} \,dr_{1} \biggr)^{\frac{1}{p_{1}}} \end{aligned}$$

where

$$ H^{*}_{1}(p_{1})= \bigl[(x) (y) (z) ( \omega ) \bigr]^{\frac{p_{1}-1}{p_{1}}}. $$

Taking \(\mathbb{T}_{1}=\mathbb{T}_{1}= \mathbb{Z}\) in Theorem 2.9, we have \(\sigma (x)=x+1\), \(\sigma (y)=y+1\), \(\sigma (\omega)=\omega+1\), \(\sigma (z)=z+1\), and we get the following result.

Corollary 2.15

Let \(\{a_{m_{1},n_{1}}\}_{0 \leqslant m_{1},n_{1} \leqslant N}\) and \(\{b_{k_{1},r_{1}}\}_{0 \leqslant k_{1},r_{1} \leqslant N}\) be nonnegative sequences of real numbers, and let \(\{p_{m_{1},n_{1}}\}_{0 \leqslant m_{1},n_{1} \leqslant N}\) and \(\{q_{k_{1},r_{1}}\}_{0 \leqslant k_{1},r_{1} \leqslant N}\) be positive sequences. Define

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

Then

$$\begin{aligned}& \sum_{s_{1}=1}^{m_{1}} \sum_{t_{1}=1}^{n_{1}} \Biggl(\sum _{k_{1}=1}^{z_{1}} \sum_{r_{1}=1}^{\omega _{1}} \frac{P_{m_{1},n_{1}} Q_{k_{1},r_{1}} \Phi ( A_{s_{1},t_{1}}) \Psi (B_{k_{1},r_{1}})}{ ( \vert h(s_{1} t_{1}) \vert ^{\frac{1}{2\beta }}+ \vert h^{*}(k_{1} r_{1}) \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Biggr) \\& \quad \leqslant H^{**}(p_{1}) \Biggl\{ \sum _{s_{1}=1}^{m_{1}} \sum_{t_{1}=1}^{n_{1}} \bigl((m_{1}+1)-s_{1}\bigr) \bigl((n_{1}+1)-t_{1} \bigr) \bigl(p_{s_{1},t_{1}} \Phi (a_{s_{1},t_{1}}) \bigr)^{p_{1}} \Biggr\} ^{\frac{1}{p_{1}}} \\& \qquad {} \times \Biggl\{ \sum_{k_{1}=1}^{z_{1}} \sum _{r_{1}=1}^{\omega _{1}}\bigl((z_{1}+1)-k_{1} \bigr) \bigl(( \omega _{1}+1)-r_{1}\bigr) \bigl( q_{k_{1},r_{1}} \Psi ( b_{k_{1},r_{1}}) \bigr)^{p_{1}} \Biggr\} ^{\frac{1}{p_{1}}}, \end{aligned}$$

where

$$ H^{**}(p_{1})=(m_{1} n_{1} z_{1} \omega _{1})^{\frac{p_{1}-1}{p_{1}}}. $$

Corollary 2.16

Under the assumptions of Theorem 2.9, we have

$$\begin{aligned}& \int _{t_{0}}^{x} \int _{t_{0}}^{y} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega } \frac{\Phi ( F (s_{1},t_{1}))\Psi ( G (k_{1},r_{1}))P (s_{1},t_{1})Q (k_{1},r_{1})}{ ( \vert h [(s_{1}-t_{0})(t_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }}+ \vert h^{*} [(k_{1}-t_{0})(r_{1}-t_{0}) ] \vert ^{\frac{1}{2 \beta }} )^{\frac{2 \alpha (p_{1}-1)}{p_{1}}}} \Delta k_{1} \Delta r_{1} \biggr)\Delta s_{1} \Delta t_{1} \\& \quad \leqslant H_{1}(p_{1}) \biggl\{ h \biggl( \int _{t_{0}}^{x} \int _{t_{0}}^{y} \bigl(\sigma (x)-s_{1} \bigr) \bigl(\sigma (y)-t_{1}\bigr) \bigl(p (s_{1},t_{1}) \Phi \bigl(f ( s_{1} ,t_{1} )\bigr) \bigr)^{p_{1}} \Delta s_{1} \Delta t_{1} \biggr) \\& \qquad {} +h^{*} \biggl( \int _{t_{0}}^{z} \int _{t_{0}}^{\omega }\bigl(\sigma (z)-k_{1} \bigr) \bigl( \sigma (\omega )-r_{1}\bigr) \bigl(q (k_{1},r_{1}) \Psi \bigl(g (k_{1} ,r_{1} )\bigr) \bigr)^{p_{1}} \Delta k_{1} \Delta r_{1} \biggr) \biggr\} ^{\frac{1}{p_{1}}}. \end{aligned}$$

Proof

Using (1.7) in (2.31), we get the desired result.. □

Conclusion

In this paper, we established some dynamic Hilbert-type inequalities in two separate variables on time scales by using the Fenchel–Legendre transform. We also applied our inequalities to discrete and continuous calculus to obtain some new inequalities as particular cases.

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Acknowledgements

The authors would like to thank the editor and reviewers for their valuable comments, which improved the paper.

Funding

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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El-Deeb, A.A., Rashid, S., Khan, Z.A. et al. New dynamic Hilbert-type inequalities in two independent variables involving Fenchel–Legendre transform. Adv Differ Equ 2021, 239 (2021). https://doi.org/10.1186/s13662-021-03394-w

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MSC

  • 26D10
  • 26D15
  • 26E70
  • 34A40

Keywords

  • Hilbert’s inequality
  • Dynamic inequality
  • Time scales
  • Fenchel–Young inequality