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.. □