1 Introduction

In recent years, Hilbert’s dual-series inequality and its integral form [1, pp. 253–254] have been granted significant attention by many scholars (for example, see [2–10]). In particular, B. G. Pachpatte [11] established a new inequality close to that of Hilbert as follows. Let \(k, r\geq 1\), \(A_{s}=\sum_{m=1}^{s}a_{m}\geq 0\) and \(B_{\vartheta }=\sum_{n=1}^{\vartheta }b_{n}\geq 0\). Then

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

where

$$ C(k, r, p, q)=\frac{1}{2}kr\sqrt{pq}. $$

In the same article [11], Pachpatte demonstrated the integral version of (1) as follows. Let \(k,r\geq 1\), \(\Pi (s)=\int _{0}^{s}\omega _{1}(\xi )\,d\xi \geq 0\) and \(\Omega (\vartheta )=\int _{0}^{\vartheta }\omega _{2}(\nu )\,d\nu \geq 0\), for \(s,\xi \in (0, x)\) and \(\vartheta ,\nu \in (0,y)\). Then

$$\begin{aligned} \int _{0}^{x} \int _{0}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{s+\vartheta }\,ds\,d \vartheta \leq &C^{\ast }(k, r, x, y) \biggl( \int _{0}^{x}(x-s) \bigl( \Pi ^{k-1}(s)\omega _{1}(s) \bigr) ^{2}\,ds \biggr) ^{\frac{1}{2}} \\ &{}\times \biggl( \int _{0}^{y}(y-\vartheta ) \bigl( \Omega ^{r-1}( \vartheta )\omega _{2}(\vartheta ) \bigr) ^{2}\,d\vartheta \biggr) ^{ \frac{1}{2}}, \end{aligned}$$
(2)

where

$$ C^{\ast }(k, r, x, y)=\frac{1}{2}kr\sqrt{xy}. $$

In [12], Young-Ho Kim gave some generalizations of (1) and (2) by introducing a parameter \(\gamma >0\) as follows. Let \(k, r\geq 1\), \(A_{s}=\sum_{m=1}^{s}a_{m}\geq 0\) and \(B_{\vartheta }=\sum_{n=1}^{\vartheta }b_{n}\geq 0\). Then

$$\begin{aligned} \sum_{s=1}^{p}\sum _{\vartheta =1}^{q} \frac{A_{s}^{k}B_{\vartheta }^{r}}{ ( s^{\gamma }+\vartheta ^{\gamma } ) ^{\frac{1}{\gamma }}} \leq &D(k, r, \gamma , p, q) \Biggl( \sum_{s=1}^{p}(p-s+1) \bigl(A_{s}^{k-1}a_{s}\bigr)^{2} \Biggr) ^{\frac{1}{2}} \\ &{}\times \Biggl( \sum_{\vartheta =1}^{q}(q- \vartheta +1) \bigl(B_{ \vartheta }^{r-1}b_{\vartheta } \bigr)^{2} \Biggr) ^{\frac{1}{2}}, \end{aligned}$$
(3)

where

$$ D(k, r, \gamma , p, q)=\biggl(\frac{1}{2}\biggr)^{ \frac{1}{\gamma }}kr \sqrt{pq}. $$

The integral version of (3) is established in the next consequence. Let \(k, r\geq 1\), \(\gamma >0\), \(\Pi (s)=\int _{0}^{s}\omega _{1}(\xi )\,d\xi \geq 0\), and \(\Omega (\vartheta )=\int _{0}^{\vartheta }\omega _{2}(\nu )\,d\nu \geq 0\), for \(s, \xi \in (0, x)\) and ϑ, \(\nu \in (0, y)\). Then

$$\begin{aligned} \int _{0}^{x} \int _{0}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{ ( s^{\gamma }+\vartheta ^{\gamma } ) ^{\frac{1}{\gamma }}}\,ds\,d \vartheta \leq &D^{\ast }(k, r, \gamma , x, y) \biggl( \int _{0}^{x}(x-s) \bigl( \Pi ^{k-1}(s)\omega _{1}(s) \bigr) ^{2}\,ds \biggr) ^{\frac{1}{2}} \\ &{}\times \biggl( \int _{0}^{y}(y-\vartheta ) \bigl( \Omega ^{r-1}( \vartheta )\omega _{2}(\vartheta ) \bigr) ^{2}\,d\vartheta \biggr) ^{ \frac{1}{2}}, \end{aligned}$$
(4)

where

$$ D^{\ast }(k, r, \gamma , x, y)=\biggl(\frac{1}{2} \biggr)^{\frac{1}{\gamma }}kr\sqrt{xy}. $$

Another refinement of inequalities (1) and (2) has been made by Yang [13] as follows. Let \(k, r\geq 1\) and \(\lambda , \mu >1\) be constants such that \(1/\lambda +1/\mu =1\), \(A_{s}=\sum_{m=1}^{s}a_{m}\geq 0\), and \(B_{\vartheta }=\sum_{n=1}^{\vartheta }b_{n}\geq 0\). Then

$$\begin{aligned} \sum_{s=1}^{p}\sum _{\vartheta =1}^{q} \frac{A_{s}^{k}B_{\vartheta }^{r}}{\mu s^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda \vartheta ^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \leq& E(k, r, \lambda , \mu , p, q) \Biggl( \sum _{s=1}^{p}(p-s+1) \bigl(A_{s}^{k-1}a_{s} \bigr)^{\lambda } \Biggr) ^{ \frac{1}{\lambda }} \\ & {}\times \Biggl( \sum_{\vartheta =1}^{q}(q- \vartheta +1) \bigl(B_{ \vartheta }^{r-1}b_{\vartheta } \bigr)^{\mu } \Biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(5)

where

$$ E(k, r, \lambda , \mu , p, q)= \frac{kr}{\lambda +\mu }p^{\frac{\lambda -1}{\lambda }}q^{\frac{\mu -1}{\mu }}. $$

The integral version of (5) is established in the next consequence. Let \(k, r\geq 1\) and \(\lambda , \mu >1\) be constants such that \(1/\lambda +1/\mu =1\), \(\Pi (s)=\int _{0}^{s}\omega _{1}(\xi )\,d\xi \geq 0\), and \(\Omega (\vartheta )=\int _{0}^{\vartheta }\omega _{2}(\nu )\,d\nu \geq 0\), for \(s, \xi \in (0, x)\) and \(\vartheta , \nu \in (0, y)\). Then

$$\begin{aligned}& \int _{0}^{x} \int _{0}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{\mu s^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda \vartheta ^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\,ds\,d \vartheta \\& \quad \leq E^{\ast }(k, r, \lambda , \mu , x, y) \biggl( \int _{0}^{x}(x-s) \bigl( \Pi ^{k-1}(s)\omega _{1}(s) \bigr) ^{\lambda }\,ds \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{0}^{y}(y-\vartheta ) \bigl( \Omega ^{r-1}( \vartheta )\omega _{2}(\vartheta ) \bigr) ^{\mu }\,d\vartheta \biggr) ^{ \frac{1}{\mu }}, \end{aligned}$$
(6)

where

$$ E^{\ast }(k, r, \lambda , \mu , x, y)=\frac{kr}{\lambda +\mu }x^{\frac{\lambda -1}{\lambda }}y^{ \frac{\mu -1}{\mu }}. $$

After construction of time scale calculus, dynamic inequalities have become the focus of interest, and classical inequalities have been established for any time scale \(\mathbb{T}\). We can refer two surveys [14, 15] and a monograph [16] for exhibition of these results.

In [17] the researchers concluded some generalizations of inequalities (1) and (2) for time scale delta calculus. Specifically, they proved that if \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\), \(\omega _{1}(s)\in \mathrm{C}_{rd}([\vartheta _{0}, x)_{\mathbb{T}}, \mathbb{R}^{+})\), \(\omega _{2}(\vartheta )\in \mathrm{C}_{rd}([\vartheta _{0}, y)_{\mathbb{T}}, \mathbb{R}^{+})\), \(k, r\geq 1\) and \(\lambda , \mu >1\) are constants such that \(1/\lambda +1/\mu =1\), then for \(s\in {}[ \vartheta _{0}, x)_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, y)_{\mathbb{T}}\), one has

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{\mu (s-\vartheta _{0})^{{\frac{1}{\lambda }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{1}{\mu }}}} \Delta s \Delta \vartheta \\& \quad \leq G(k, r, \lambda , \mu , x, y) \biggl( \int _{\vartheta _{0}}^{x}\bigl(\sigma (x)-s\bigr) \bigl( \Pi ^{k-1}\bigl( \sigma (s)\bigr)\omega _{1}(s) \bigr) ^{\lambda }\Delta s \biggr) ^{ \frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(\sigma (y)-\vartheta \bigr) \bigl( \Omega ^{r-1}\bigl(\sigma (\vartheta )\bigr)\omega _{2}(\vartheta ) \bigr) ^{\mu }\Delta \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(7)

where \(\Pi (s)=\int _{\vartheta _{0}}^{s}\omega _{1}(\xi )\Delta \xi \), \(\Omega (\vartheta )=\int _{\vartheta _{0}}^{\vartheta }\omega _{2}( \xi )\Delta \xi \), and

$$ G(k, r, \lambda , \mu , x, y)= \frac{kr}{\lambda \mu }(x-\vartheta _{0})^{\frac{\lambda -1}{\lambda }}(y- \vartheta _{0})^{\frac{\mu -1}{\mu }}. $$
(8)

Another refinement of (7) for time scale delta calculus has been made by Rezk et al. [18] as follows. Let \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\), \(\omega _{1}(s)\in \mathrm{C}_{rd}([\vartheta _{0}, x)_{\mathbb{T}}, \mathbb{R}^{+})\), \(\omega _{2}(\vartheta )\in \mathrm{C}_{rd}([\vartheta _{0}, y)_{\mathbb{T}}, \mathbb{R}^{+})\), \(k, r\geq 1\) and \(\lambda , \mu >1\) be constants such that \(1/\lambda +1/\mu =1\), then for \(s\in {}[ \vartheta _{0}, \rho )_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, \tau )_{\mathbb{T}}\), one has

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{\mu (s-\vartheta _{0})^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda (\vartheta -\vartheta _{0})^{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}} \Delta s \Delta \vartheta \\& \quad \leq G^{\ast }(k, r, \lambda , \mu , x, y) \biggl( \int _{\vartheta _{0}}^{x}\bigl(\sigma (x)-s\bigr) \bigl( \Pi ^{k-1}\bigl( \sigma (s)\bigr)\omega _{1}(s) \bigr) ^{\lambda }\Delta s \biggr) ^{ \frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(\sigma (y)-\vartheta \bigr) \bigl( \Omega ^{r-1}\bigl(\sigma (\vartheta )\bigr)\omega _{2}(\vartheta ) \bigr) ^{\mu }\Delta \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(9)

where \(\Pi (s)=\int _{\vartheta _{0}}^{s}\omega _{1}(\xi )\Delta \xi \), \(\Omega (\vartheta )=\int _{\vartheta _{0}}^{\vartheta }\omega _{2}( \xi )\Delta \xi \), and

$$ G^{\ast }(k, r, \lambda , \mu , x, y)=\frac{kr}{\lambda +\mu }(x- \vartheta _{0})^{ \frac{\lambda -1}{\lambda }}(y-\vartheta _{0})^{\frac{\mu -1}{\mu }}. $$
(10)

For developing of Hilbert’s inequalities for time scale delta calculus, we refer the reader to the articles [19–29]. Although there are many results for time scale calculus in the sense of delta derivative, there is not much done for the nabla derivative. Therefore the major contribution of this article is to extend Hilbert-type inequalities for the nabla time scale calculus and to unify them for an arbitrary time scale. The main theorems are inspired from the paper [18] which presents the corresponding results for time scale delta calculus. By obtaining their nabla versions, we can show the generalizations of these inequalities for different types of time scales \(\mathbb{T}\), such as real numbers and integers.

The structure of this paper can be listed as follows. Section 2 presents the fundamental concepts of the time scale calculus in terms of delta and nabla derivatives. Section 3 is devoted to main results, which are to generalize inequalities (5) and (6) for the nabla time scale calculus and so, to obtain nabla calculus versions of (9) and several inequalities of Hilbert’s type in [18].

2 Preliminaries

In this section, the fundamental theories of the time scale delta and nabla calculi will be presented. Time scale calculus whose detailed information can be found in [30, 31] has been invented in order to unify continuous and discrete analysis.

A nonempty closed subset of \(\mathbb{R}\) is named a time scale and is denoted by \(\mathbb{T}\). For \(\vartheta \in \mathbb{T}\), if \(\inf \emptyset =\sup \mathbb{T}\) and \(\sup \emptyset =\inf \mathbb{T}\), then the forward jump operator \(\sigma :\mathbb{T}\rightarrow \mathbb{T}\) and the backward jump operator \(\rho :\mathbb{T}\rightarrow \mathbb{T}\) are defined as \(\sigma (\vartheta )=\inf (\vartheta , \infty )_{\mathbb{T}}\) and \(\rho (\vartheta )=\sup (-\infty , \vartheta )_{\mathbb{T}}\), respectively. From the above two concepts, it can be mentioned that a point \(\vartheta \in \mathbb{T}\) with \(\inf \mathbb{T}<\vartheta <\sup \mathbb{T}\) is named right-scattered if \(\sigma (\vartheta )>\vartheta \), right-dense if \(\sigma (\vartheta )=\vartheta \), left-scattered if \(\rho (\vartheta )<\vartheta \) and left-dense if \(\rho (\vartheta )=\vartheta \).

The Δ-derivative of \(\psi :\mathbb{T}\rightarrow \mathbb{R}\) at \(\vartheta \in \mathbb{T}^{k}=\mathbb{T}/(\rho (\sup \mathbb{T}), \sup \mathbb{T]}\) denoted by \(\psi ^{\Delta }(\vartheta )\) is the number enjoying the property that for all \(\varepsilon >0\) there is a neighborhood U of \(\vartheta \in \mathbb{T}^{k}\) such that

$$ \bigl\vert \psi \bigl(\sigma (\vartheta )\bigr)-\psi (s)-\psi ^{\Delta }( \vartheta ) \bigl(\sigma (\vartheta )-s\bigr) \bigr\vert \leq \varepsilon \bigl\vert \sigma (\vartheta )-s \bigr\vert ,\quad \text{for all }s \in U. $$

The ∇-derivative of \(\psi :\mathbb{T}\rightarrow \mathbb{R}\) at \(\vartheta \in \mathbb{T}_{k}=\mathbb{T}/[\inf \mathbb{T}, \sigma (\inf \mathbb{T}))\) denoted by \(\psi ^{\nabla }(\xi )\) is the number enjoying the property that for all \(\varepsilon >0\) there is a neighborhood V of \(\vartheta \in \mathbb{T}_{k}\) such that

$$ \bigl\vert \psi (\vartheta )-\psi \bigl(\rho (s)\bigr)-\psi ^{\nabla }( \vartheta ) \bigl(\vartheta -\rho (s)\bigr) \bigr\vert \leq \varepsilon \bigl\vert s-\rho (\vartheta ) \bigr\vert ,\quad \text{for all }s\in V. $$

A function \(\psi :\mathbb{T}\rightarrow \mathbb{R}\) is rd-continuous if it is continuous at each right-dense point in \(\mathbb{T}\) and \(\underset{s\rightarrow \vartheta ^{-}}{\lim }\psi (s)\) exists as a finite number for all left-dense points in \(\mathbb{T}\). The set \(\mathrm{C}_{rd}(\mathbb{T}, {\mathbb{R)}}\) represents the class of real, rd-continuous functions defined on \(\mathbb{T}\). If \(\psi \in \mathrm{C}_{rd}(\mathbb{T}, {\mathbb{R)}}\), then there exists a function \(\Psi (\vartheta )\) such that \(\Psi ^{\Delta }(\vartheta )=\psi (\vartheta )\) and the delta integral of ψ is defined by

$$ \int _{x_{0}}^{x}\psi (\vartheta )\Delta \vartheta =\Psi (x)-\Psi (x_{0}). $$

A function \(\psi :\mathbb{T}\rightarrow \mathbb{R}\) is ld-continuous if it is continuous at each left-dense point in \(\mathbb{T}\) and \(\underset{s\rightarrow \vartheta ^{+}}{\lim }\psi (s)\) exists as a finite number for all right-dense points in \(\mathbb{T}\). The set \(\mathrm{C}_{ld}(\mathbb{T}, {\mathbb{R)}}\) represents the class of real, ld-continuous functions defined on \(\mathbb{T}\). If ψ∈ \(\mathrm{C}_{ld}(\mathbb{T}, {\mathbb{R)}}\), then there exists a function \(\Psi (\vartheta )\) such that \(\Psi ^{\nabla }(\vartheta )=\psi (\vartheta )\) and the nabla integral of ψ is defined by

$$ \int _{x_{0}}^{x}\psi (\vartheta )\nabla \vartheta =\Psi (x)-\Psi (x_{0}). $$

In the following, we display some basic lemmas and algebraic inequalities that play a key role in proving the major findings of this paper.

Lemma 2.1

(Nabla Hölder’s Inequality [32])

Let \(x_{0}, x\in \mathbb{T}\). For \(\xi , \psi \in \mathrm{C}_{ld}([x_{0}, x]_{\mathbb{T}}, \mathbb{R})\), we have

$$ \int _{x_{0}}^{x}\xi (\vartheta )\psi (\vartheta ) \nabla \vartheta \leq \biggl( \int _{x_{0}}^{x}\xi ^{\lambda }(\vartheta ) \nabla \vartheta \biggr) ^{\frac{1}{\lambda }} \biggl( \int _{x_{0}}^{x}\psi ^{\mu }(\vartheta ) \nabla \vartheta \biggr) ^{\frac{1}{\mu }}, $$
(11)

where \(\lambda , \mu >1\) with \(1/\lambda +1/\mu =1\).

Lemma 2.2

(Nabla Jensen’s Inequality [33, Theorem 3.4])

Let \(x_{0}\), \(x\in \mathbb{T}\) and \(m, n\in \mathbb{R}\). Assume that \(\xi \in \mathrm{C}_{ld}([x_{0}, x]_{ \mathbb{T}}, (m, n))\) and \(\psi \in \mathrm{C}_{ld}([x_{0}, x]_{\mathbb{T}} , \mathbb{R})\) are nonnegative with \(\int _{x_{0}}^{x} \xi (\eta )\Delta \eta >0\). If \(\Theta \in \mathrm{C}((m, n), \mathbb{R})\) is a convex function, then

$$ \Theta \biggl( \frac{\int _{x_{0}}^{x}\xi (\eta )\psi (\eta )\nabla \eta }{\int _{x_{0}}^{x}\xi (\eta )\nabla \eta } \biggr) \leq \frac{\int _{x_{0}}^{x}\xi (\eta )\Theta (\psi (\eta ))\nabla \eta }{\int _{x_{0}}^{x}\xi (\eta )\nabla \eta }. $$
(12)

Lemma 2.3

(The power rule for nabla derivative [33, Lemma 3.1])

Let \(x_{0}\), \(x\in \mathbb{T,}\) \(\psi \in \mathrm{C}_{ld}([x_{0}, x]_{\mathbb{T}}, \mathbb{R})\) be a nonnegative function, and \(\gamma \geq 1\) a real constant. Then

$$ \biggl( \int _{x_{0}}^{x}\psi (\xi )\nabla \xi \biggr) ^{\gamma } \leq \gamma \int _{x_{0}}^{x}\psi (v) \biggl( \int _{a}^{v}\psi (\xi ) \nabla \xi \biggr) ^{\gamma -1}\nabla v. $$
(13)

Lemma 2.4

(Young’s inequality [34])

Let \(\delta >0\), \(\Lambda _{q}>0\) and \(\sum_{q=1}^{n}\Lambda _{q}=\Upsilon _{n}\). Then

$$ \Biggl\{ \prod_{q=1}^{n}s_{q}^{\Lambda _{q}} \Biggr\} ^{ \frac{1}{\Upsilon _{n}}}\leq \Biggl\{ \frac{1}{\Upsilon _{n}}\sum _{q=1}^{n} \Lambda _{q}s_{q}^{\delta } \Biggr\} ^{\frac{1}{\delta }}. $$
(14)

Lemma 2.5

([33, Lemma 3.2])

Let s, ϑ, \(\vartheta _{0}\in \mathbb{T}\) with \(s, \vartheta \geq \vartheta _{0}\) and \(\psi \in \mathrm{C}_{ld}([a, b]_{\mathbb{T}}, \mathbb{R})\). Then

$$ \int _{\vartheta _{0}}^{s} \biggl( \int _{\vartheta _{0}}^{\vartheta } \psi (\xi )\nabla \xi \biggr) \nabla \vartheta = \int _{\vartheta _{0}}^{s} \biggl( \int _{\rho (\xi )}^{s}\psi (\xi )\nabla s \biggr) \nabla \xi = \int _{\vartheta _{0}}^{s} \bigl( s-\rho (\xi ) \bigr) \psi ( \xi )\nabla \xi . $$
(15)

3 Key results

In this section, we focus on obtaining the corresponding outcomes for the nabla time scale calculation in [18]. We must assume that all functions found in the theorem statements are nonnegative, ld-continuous, ∇-differentiable, and locally nabla integrable.

Theorem 3.1

Let \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\) and \(\omega _{1}\in \mathrm{C}_{ld}([\vartheta _{0}, x]_{\mathbb{T}}, \mathbb{R}^{+})\), \(\omega _{2}\in \mathrm{C}_{ld}([\vartheta _{0}, y]_{\mathbb{T}}, \mathbb{R}^{+})\). Define

$$ \Pi (s)= \int _{\vartheta _{0}}^{s}\omega _{1}(\xi ) \nabla \xi \quad \textit{and}\quad \Omega (\vartheta )= \int _{\vartheta _{0}}^{ \vartheta }\omega _{2}(\xi ) \nabla \xi . $$
(16)

Then for \(s\in {}[ \vartheta _{0}, x]_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, y]_{\mathbb{T}}\), we have

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq H(k, r, \lambda , \mu , x, y) \biggl( \int _{\vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \bigl( \Pi ^{k-1}(s) \omega _{1}(s) \bigr) ^{\lambda }\nabla s \biggr) ^{ \frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl( \Omega ^{r-1}(\vartheta )\omega _{2}(\vartheta ) \bigr) ^{ \mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(17)

where

$$ H(k, r, \lambda , \mu , x, y)= \frac{kr}{\lambda +\mu }(x-\vartheta _{0})^{\frac{\lambda -1}{\lambda }}(y- \vartheta _{0})^{\frac{\mu -1}{\mu }}. $$

Proof

By using (13), we obtain

$$ \Pi ^{k}(s)\leq k \int _{\vartheta _{0}}^{s}\Pi ^{k-1}(\xi )\omega _{1}( \xi )\nabla \xi $$
(18)

and

$$ \Omega ^{r}(\vartheta )\leq r \int _{\vartheta _{0}}^{\vartheta } \Omega ^{r-1}(\xi ) \omega _{2}(\xi )\nabla \xi . $$
(19)

Then, we have

$$ \Pi ^{k}(s)\Omega ^{r}(\vartheta )\leq kr \biggl( \int _{\vartheta _{0}}^{s} \Pi ^{k-1}(\xi )\omega _{1}(\xi )\nabla \xi \biggr) \biggl( \int _{ \vartheta _{0}}^{\vartheta }\Omega ^{r-1}(\xi ) \omega _{2}(\xi ) \nabla \xi \biggr) . $$
(20)

Applying (11) to \(\int _{\vartheta _{0}}^{s}\Pi ^{k-1}(\xi )\omega _{1}(\xi )\nabla \xi \) with indices λ and \(\lambda /(\lambda -1)\), we find that

$$ \int _{\vartheta _{0}}^{s}\Pi ^{k-1}(\xi )\omega _{1}(\xi )\nabla \xi \leq (s-\vartheta _{0})^{\frac{\lambda -1}{\lambda }} \biggl( \int _{\vartheta _{0}}^{s} \bigl( \Pi ^{k-1}(\xi ) \omega _{1}(\xi ) \bigr) ^{\lambda }\nabla \xi \biggr) ^{\frac{1}{\lambda }}, $$
(21)

while doing the same to the integral \(\int _{\vartheta _{0}}^{\vartheta }\Omega ^{r-1}(\xi )\omega _{2}( \xi )\nabla \xi \) with indices μ and \(\mu /(\mu -1)\), we find that

$$ \int _{\vartheta _{0}}^{\vartheta }\Omega ^{r-1}(\xi ) \omega _{2}( \xi )\nabla \xi \leq (\vartheta -\vartheta _{0})^{\frac{\mu -1}{\mu }} \biggl( \int _{\vartheta _{0}}^{\vartheta } \bigl( \Omega ^{r-1}( \xi ) \omega _{2}(\xi ) \bigr) ^{\mu }\nabla \xi \biggr) ^{\frac{1}{\mu }}. $$
(22)

From (20), (21), and (22), we get

$$\begin{aligned} \Pi ^{k}(s)\Omega ^{r}(\vartheta ) \leq &kr(s-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }}(\vartheta -\vartheta _{0})^{ \frac{\mu -1}{\mu }} \biggl( \int _{\vartheta _{0}}^{s} \bigl( \Pi ^{k-1}(\xi ) \omega _{1}( \xi ) \bigr) ^{\lambda }\nabla \xi \biggr) ^{\frac{1}{\lambda }} \\ &{}\times \biggl( \int _{\vartheta _{0}}^{\vartheta } \bigl( \Omega ^{r-1}( \xi )\omega _{2}(\xi ) \bigr) ^{\mu }\nabla \xi \biggr) ^{ \frac{1}{\mu }}. \end{aligned}$$
(23)

Using inequality (14), we note

$$ \bigl( s_{1}^{\Lambda _{1}}s_{2}^{\Lambda _{2}} \bigr) ^{ \frac{\delta }{\Lambda _{1}+\Lambda _{2}}}\leq \frac{1}{\Lambda _{1}+\Lambda _{2}} \bigl( \Lambda _{1}s_{1}^{\delta }+\Lambda _{2}s_{2}^{\delta } \bigr) . $$
(24)

Now, by setting \(s_{1}=(s-\vartheta _{0})^{\lambda -1}\), \(s_{2}=(\vartheta -\vartheta _{0})^{\mu -1}\), \(\Lambda _{1}=1/\lambda \), \(\Lambda _{1}=1/\mu \), and \(\delta =\Lambda _{1}+\Lambda _{2}\) in (24), we get

$$ (s-\vartheta _{0})^{\frac{\lambda -1}{\lambda }}(\vartheta - \vartheta _{0})^{\frac{\mu -1}{\mu }}\leq \frac{\lambda \mu }{\lambda +\mu } \biggl( \frac{(s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}{\lambda }+ \frac{(\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\mu } \biggr) . $$
(25)

Substituting (25) into (23) yields

$$\begin{aligned} \Pi ^{k}(s)\Omega ^{r}(\vartheta ) \leq & \frac{kr\lambda \mu }{\lambda +\mu } \biggl( \frac{(s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}{\lambda }+ \frac{(\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\mu } \biggr) \\ &{}\times \biggl( \int _{\vartheta _{0}}^{s} \bigl( \Pi ^{k-1}(\xi ) \omega _{1}(\xi ) \bigr) ^{\lambda }\nabla \xi \biggr) ^{ \frac{1}{\lambda }} \biggl( \int _{\vartheta _{0}}^{\vartheta } \bigl( \Omega ^{r-1}( \xi )\omega _{2}(\xi ) \bigr) ^{\mu }\nabla \xi \biggr) ^{\frac{1}{\mu }}. \end{aligned}$$
(26)

Dividing both sides of (26) by \(\mu (s-\vartheta _{0})^{ [ (\lambda -1)(\lambda +\mu ) ] / \lambda \mu }+\lambda (\vartheta -\vartheta _{0})^{{ [ (\mu -1)( \lambda +\mu ) ] /\lambda \mu }}\), we obtain

$$\begin{aligned}& \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \\& \quad \leq \frac{kr}{\lambda +\mu } \biggl( \int _{\vartheta _{0}}^{s} \bigl( \Pi ^{k-1}(\xi ) \omega _{1}(\xi ) \bigr) ^{\lambda }\nabla \xi \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{\vartheta } \bigl( \Omega ^{r-1}( \xi )\omega _{2}(\xi ) \bigr) ^{\mu }\nabla \xi \biggr) ^{ \frac{1}{\mu }}. \end{aligned}$$
(27)

Integrating both sides of (27) and using (11) again, we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq \frac{kr}{\lambda +\mu }(x-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }}(y- \vartheta _{0})^{\frac{\mu -1}{\mu }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{x} \biggl( \int _{\vartheta _{0}}^{s} \bigl( \Pi ^{k-1}(\xi ) \omega _{1}(\xi ) \bigr) ^{\lambda }\nabla \xi \biggr) \nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y} \biggl( \int _{\vartheta _{0}}^{ \vartheta } \bigl( \Omega ^{r-1}( \xi )\omega _{2}(\xi ) \bigr) ^{ \mu }\nabla \xi \biggr) \nabla \vartheta \biggr) ^{\frac{1}{\mu }}. \end{aligned}$$
(28)

Applying Lemma 2.5 on (28), we conclude that

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq \frac{kr}{\lambda +\mu }(x-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }}(y- \vartheta _{0})^{\frac{\mu -1}{\mu }} \biggl( \int _{\vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \bigl( \Pi ^{k-1}(s) \omega _{1}(s) \bigr) ^{\lambda }\nabla s \biggr) ^{ \frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl( \Omega ^{r-1}(\vartheta )\omega _{2}(\vartheta ) \bigr) ^{ \mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }} \\& \quad =H(k, r, \lambda , \mu , x, y) \biggl( \int _{\vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \bigl( \Pi ^{k-1}(s)\omega _{1}(s) \bigr) ^{\lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl( \Omega ^{r-1}(\vartheta )\omega _{2}(\vartheta ) \bigr) ^{ \mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$

that is, (17) is true. □

Remark 3.2

By setting \(1/\lambda +1/\mu =1\) in (24), we obtain

$$ \bigl( s_{1}^{\Lambda _{1}}s_{2}^{\Lambda _{2}} \bigr) \leq \frac{1}{\Lambda _{1}+\Lambda _{2}} \bigl( \Lambda _{1}s_{1}^{ \Lambda _{1}+\Lambda _{2}}+ \Lambda _{2}s_{2}^{\Lambda _{1}+\Lambda _{2}} \bigr) . $$
(29)

Hence, by applying (29) on the right-hand side of (17) in Theorem 3.1, we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq \frac{\lambda \mu kr}{ ( \lambda +\mu ) ^{2}}(x- \vartheta _{0})^{\frac{\lambda -1}{\lambda }}(y- \vartheta _{0})^{ \frac{\mu -1}{\mu }} \\& \qquad {}\times \biggl\{ \frac{1}{\lambda } \biggl( \int _{\vartheta _{0}}^{x}\bigl(x- \rho (s)\bigr) \bigl( \Pi ^{k-1}(s)\omega _{1}(s) \bigr) ^{\lambda } \nabla s \biggr) ^{\frac{\lambda +\mu }{\lambda \mu }} \\& \qquad {} +\frac{1}{\mu } \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho ( \vartheta )\bigr) \bigl( \Omega ^{r-1}(\vartheta )\omega _{2}(\vartheta ) \bigr) ^{\mu }\nabla \vartheta \biggr) ^{ \frac{\lambda +\mu }{\lambda \mu }} \biggr\} . \end{aligned}$$

Corollary 3.1

If we take \(1/\lambda +1/\mu =1\) in (17), then

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{\mu (s-\vartheta _{0})^{{\lambda -1}}+\lambda (\vartheta -\vartheta _{0})^{{\mu -1}}} \nabla s \nabla \vartheta \\& \quad \leq H^{\ast }(k, r, \lambda , \mu , x, y) \biggl( \int _{\vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \bigl( \Pi ^{k-1}(s) \omega _{1}(s) \bigr) ^{\lambda }\nabla s \biggr) ^{ \frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl( \Omega ^{r-1}(\vartheta )\omega _{2}(\vartheta ) \bigr) ^{ \mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(30)

where

$$ H^{\ast }(k, r, \lambda , \mu , x, y)=\frac{kr}{\lambda \mu }(x- \vartheta _{0})^{ \frac{\lambda -1}{\lambda }}(y-\vartheta _{0})^{\frac{\mu -1}{\mu }}. $$

Remark 3.3

As a particular case of Corollary 3.1, if \(\lambda =\mu =2\), then we have

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Pi ^{k}(s)\Omega ^{r}(\vartheta )}{s+\vartheta -2\vartheta _{0}} \nabla s \nabla \vartheta \\& \quad \leq \frac{1}{2}kr \biggl( (x-\vartheta _{0}) \int _{\vartheta _{0}}^{x}\bigl(x- \rho (s)\bigr) \bigl( \Pi ^{k-1}(s)\omega _{1}(s) \bigr) ^{2}\nabla s \biggr) ^{\frac{1}{2}} \\& \qquad {}\times \biggl( (y-\vartheta _{0}) \int _{\vartheta _{0}}^{y}\bigl(y-\rho ( \vartheta )\bigr) \bigl( \Omega ^{r-1}(\vartheta )\omega _{2}(\vartheta ) \bigr) ^{2}\nabla \vartheta \biggr) ^{\frac{1}{2}}, \end{aligned}$$
(31)

which is [33, Theorem 3.3].

Remark 3.4

Clearly, for \(\mathbb{T}=\mathbb{Z}\) or \(\mathbb{T}=\mathbb{R,}\) and \(\vartheta _{0}=0\), together with \(\rho (u)=u-1\) or \(\rho (u)=u\), (17) reduces to (5) or (6), respectively.

Remark 3.5

In Theorem 3.1, if we take \(k=r=1\), then we have

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Pi (s)\Omega (\vartheta )}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq H^{\ast \ast }(\lambda , \mu , x, y) \biggl( \int _{\vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \bigl( \omega _{1}(s) \bigr) ^{ \lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl( \omega _{2}(\vartheta ) \bigr) ^{\mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(32)

where

$$ H^{\ast \ast }(\lambda , \mu , x, y)= \frac{1}{\lambda +\mu }(x-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }}(y-\vartheta _{0})^{\frac{\mu -1}{\mu }}. $$

For \(\lambda =\mu =2\), this is Anderson’s result [33, Remark 4].

In what follows, we give a further generalization of (32) obtained in Remark 3.5. Before giving our results, we presume that there are Φ and Ψ which are real-valued, nonnegative, convex and submultiplicative functions defined on \([ 0, \infty ) \). A function ψ is submultiplicative if \(\psi (s\vartheta )\leq \psi (s)\psi (\vartheta )\) for \(s, \vartheta \geq 0\).

Theorem 3.6

Let \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\) and \(\Pi (s)\), \(\Omega ( \vartheta )\) be as in Theorem 3.1and let \(k(\xi )\), \(l(\xi )\) be two positive functions defined for \(\xi \in {}[ \vartheta _{0}, x]_{\mathbb{T}}\) and \(\xi \in {}[ \vartheta _{0}, y]_{\mathbb{T}}\). Suppose that

$$ K(s)= \int _{\vartheta _{0}}^{s}k(\xi )\nabla \xi \quad \textit{and}\quad L(\vartheta )= \int _{\vartheta _{0}}^{\vartheta }l(\xi )\nabla \xi . $$
(33)

Then for \(s\in {}[ \vartheta _{0}, x]_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, y]_{\mathbb{T}}\), we have

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq M(\lambda , \mu , x, y) \biggl( \int _{ \vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \biggl( k(s)\Phi \biggl( \frac{\omega _{1}(s)}{k(s)} \biggr) \biggr) ^{\lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \biggl( l(\vartheta )\Psi \biggl( \frac{\omega _{2}(\vartheta )}{l(\vartheta )} \biggr) \biggr) ^{\mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(34)

where

$$\begin{aligned} M(\lambda ,\mu ,x,y) =&\frac{1}{\lambda +\mu } \biggl( \int _{\vartheta _{0}}^{x} \biggl( \frac{\Phi (K(s))}{K(s)} \biggr) ^{ \frac{\lambda }{\lambda -1}}\nabla s \biggr) ^{\frac{\lambda -1}{\lambda }} \\ &{}\times \biggl( \int _{ \vartheta _{0}}^{y} \biggl( \frac{\Psi (L(\vartheta ))}{L(\vartheta )} \biggr) ^{\frac{\mu }{\mu -1}}\nabla \vartheta \biggr) ^{\frac{\mu -1}{\mu }}. \end{aligned}$$

Proof

Using Jensen’s inequality (12) and the properties of Φ, we obtain

$$\begin{aligned} \Phi \bigl(\Pi (s)\bigr) =&\Phi \biggl( \frac{K(s)\int _{\vartheta _{0}}^{s}k(\xi )\frac{\omega _{1}(\xi )}{k(\xi )}\nabla \xi }{\int _{\vartheta _{0}}^{s}k(\xi )\nabla \xi } \biggr) \\ \leq &\Phi \bigl(K(s)\bigr)\Phi \biggl( \frac{\int _{\vartheta _{0}}^{s}k(\xi )\frac{\omega _{1}(\xi )}{k(\xi )}\nabla \xi }{\int _{\vartheta _{0}}^{s}k(\xi )\nabla \xi } \biggr) \\ \leq &\frac{\Phi (K(s))}{K(s)} \int _{\vartheta _{0}}^{s}k(\xi )\Phi \biggl( \frac{\omega _{1}(\xi )}{k(\xi )} \biggr) \nabla \xi . \end{aligned}$$
(35)

Further, by (11), we find that

$$ \Phi \bigl(\Pi (s)\bigr)\leq \frac{\Phi (K(s))}{K(s)}(s-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }} \biggl( \int _{\vartheta _{0}}^{s} \biggl( k( \xi )\Phi \biggl( \frac{\omega _{1}(\xi )}{k(\xi )} \biggr) \biggr) ^{ \lambda }\nabla \xi \biggr) ^{\frac{1}{\lambda }}. $$
(36)

Analogously,

$$ \Psi \bigl(\Omega (\vartheta )\bigr)\leq \frac{\Psi (L(\vartheta ))}{L(\vartheta )}( \vartheta -\vartheta _{0})^{\frac{\mu -1}{\mu }} \biggl( \int _{ \vartheta _{0}}^{\vartheta } \biggl( l(\xi )\Psi \biggl( \frac{\omega _{2}(\xi )}{l(\xi )} \biggr) \biggr) ^{\mu }\nabla \xi \biggr) ^{\frac{1}{\mu }}. $$
(37)

By multiplying (36) and (37), we get

$$\begin{aligned}& \Phi \bigl(\Pi (s)\bigr)\Psi \bigl(\Omega (\vartheta )\bigr) \\& \quad \leq (s-\vartheta _{0})^{\frac{\lambda -1}{\lambda }}(\vartheta - \vartheta _{0})^{\frac{\mu -1}{\mu }} \biggl( \frac{\Phi (K(s))}{K(s)} \biggl( \int _{\vartheta _{0}}^{s} \biggl( k(\xi )\Phi \biggl( \frac{\omega _{1}(\xi )}{k(\xi )} \biggr) \biggr) ^{\lambda }\nabla \xi \biggr) ^{ \frac{1}{\lambda }} \biggr) \\& \qquad {}\times \biggl( \frac{\Psi (L(\vartheta ))}{L(\vartheta )} \biggl( \int _{\vartheta _{0}}^{\vartheta } \biggl( l(\xi )\Psi \biggl( \frac{\omega _{2}(\xi )}{l(\xi )} \biggr) \biggr) ^{\mu }\nabla \xi \biggr) ^{\frac{1}{\mu }} \biggr) . \end{aligned}$$
(38)

Applying (24) on the term \((s-\vartheta _{0})^{(\lambda -1)/\lambda }\times (\vartheta - \vartheta _{0})^{(\mu -1)/\mu }\) gives

$$\begin{aligned}& \Phi \bigl(\Pi (s)\bigr)\Psi \bigl(\Omega (\vartheta )\bigr) \\& \quad \leq \frac{\lambda \mu }{\lambda +\mu } \biggl( \frac{(s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}{\lambda }+ \frac{(t-\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\mu } \biggr) \\& \qquad {}\times \biggl( \frac{\Phi (K(s))}{K(s)} \biggl( \int _{\vartheta _{0}}^{s} \biggl( k(\xi )\Phi \biggl( \frac{\omega _{1}(\xi )}{k(\xi )} \biggr) \biggr) ^{\lambda }\nabla \xi \biggr) ^{\frac{1}{\lambda }} \biggr) \\& \qquad {}\times \biggl( \frac{\Psi (L(\vartheta ))}{L(\vartheta )} \biggl( \int _{\vartheta _{0}}^{\vartheta } \biggl( l(\xi )\Psi \biggl( \frac{\omega _{2}(\xi )}{l(\xi )} \biggr) \biggr) ^{\mu }\nabla \xi \biggr) ^{\frac{1}{\mu }} \biggr) . \end{aligned}$$
(39)

From (39), we observe that

$$\begin{aligned}& \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \\& \quad \leq \frac{1}{\lambda +\mu } \biggl( \frac{\Phi (K(s))}{K(s)} \biggl( \int _{\vartheta _{0}}^{s} \biggl( k(\xi )\Phi \biggl( \frac{\omega _{1}(\xi )}{k(\xi )} \biggr) \biggr) ^{\lambda }\nabla \xi \biggr) ^{ \frac{1}{\lambda }} \biggr) \\& \qquad {}\times \biggl( \frac{\Psi (L(\vartheta ))}{L(\vartheta )} \biggl( \int _{\vartheta _{0}}^{\vartheta } \biggl( l(\xi )\Psi \biggl( \frac{\omega _{2}(\xi )}{l(\xi )} \biggr) \biggr) ^{\mu }\nabla \xi \biggr) ^{\frac{1}{\mu }} \biggr) . \end{aligned}$$
(40)

Integrating both sides of (40) and using (11) again with indices λ, \(\lambda /(\lambda -1)\) and μ, \(\mu /(\mu -1)\), we find that

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq \frac{1}{\lambda +\mu } \biggl( \int _{\vartheta _{0}}^{x} \biggl( \frac{\Phi (K(s))}{K(s)} \biggr) ^{\frac{\lambda }{\lambda -1}}\nabla s \biggr) ^{\frac{\lambda -1}{\lambda }} \biggl( \int _{\vartheta _{0}}^{x} \int _{ \vartheta _{0}}^{s} \biggl( k(\xi )\Phi \biggl( \frac{\omega _{1}(\xi )}{k(\xi )} \biggr) \biggr) ^{\lambda }\nabla \xi \nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y} \biggl( \frac{\Psi (L(\vartheta ))}{L(\vartheta )} \biggr) ^{\frac{\mu }{\mu -1}}\nabla \vartheta \biggr) ^{\frac{\mu -1}{\mu }} \biggl( \int _{\vartheta _{0}}^{y} \int _{\vartheta _{0}}^{ \vartheta } \biggl( l(\xi )\Psi \biggl( \frac{\omega _{2}(\xi )}{l(\xi )} \biggr) \biggr) ^{\mu }\nabla \xi \nabla \vartheta \biggr) ^{ \frac{1}{\mu }}. \end{aligned}$$
(41)

Applying Lemma 2.5 to (41), we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq \frac{1}{\lambda +\mu } \biggl( \int _{\vartheta _{0}}^{x} \biggl( \frac{\Phi (K(s))}{K(s)} \biggr) ^{\frac{\lambda }{\lambda -1}}\nabla s \biggr) ^{\frac{\lambda -1}{\lambda }} \biggl( \int _{\vartheta _{0}}^{y} \biggl( \frac{\Psi (L(\vartheta ))}{L(\vartheta )} \biggr) ^{\frac{\mu }{\mu -1}} \nabla \vartheta \biggr) ^{\frac{\mu -1}{\mu }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \biggl( k(s) \Phi \biggl( \frac{\omega _{1}(s)}{k(s)} \biggr) \biggr) ^{\lambda } \nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \biggl( l(\vartheta )\Psi \biggl( \frac{\omega _{2}(\vartheta )}{l(\vartheta )} \biggr) \biggr) ^{\mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }} \\& \quad =M(\lambda , \mu , x, y) \biggl( \int _{\vartheta _{0}}^{x}\bigl(x- \rho (s)\bigr) \biggl( k(s)\Phi \biggl( \frac{\omega _{1}(s)}{k(s)} \biggr) \biggr) ^{\lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \biggl( l(\vartheta )\Psi \biggl( \frac{\omega _{2}(\vartheta )}{l(\vartheta )} \biggr) \biggr) ^{\mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$

which is (34). □

Corollary 3.2

If we take \(1/\lambda +1/\mu =1\) in (34), then we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))}{\mu (s-\vartheta _{0})^{{\lambda -1}}+\lambda (\vartheta -\vartheta _{0})^{{\mu -1}}} \nabla s \nabla \vartheta \\& \quad \leq M^{\ast }(\lambda , \mu , x, y) \biggl( \int _{ \vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \biggl( k(s)\Phi \biggl( \frac{\omega _{1}(s)}{k(s)} \biggr) \biggr) ^{\lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \biggl( l(\vartheta )\Psi \biggl( \frac{\omega _{2}(\vartheta )}{l(\vartheta )} \biggr) \biggr) ^{\mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(42)

where

$$\begin{aligned} M^{\ast }(\lambda ,\mu ,x,y) =&\frac{1}{\lambda \mu } \biggl( \int _{ \vartheta _{0}}^{x} \biggl( \frac{\Phi (K(s))}{K(s)} \biggr) ^{ \frac{\lambda }{\lambda -1}}\nabla s \biggr) ^{\frac{\lambda -1}{\lambda }} \\ &{}\times \biggl( \int _{ \vartheta _{0}}^{y} \biggl( \frac{\Psi (L(\vartheta ))}{L(\vartheta )} \biggr) ^{\frac{\mu }{\mu -1}}\nabla \vartheta \biggr) ^{\frac{\mu -1}{\mu }}. \end{aligned}$$

Remark 3.7

As a particular case of Corollary 3.2, if \(\lambda =\mu =2\), then we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))}{s+\vartheta -2\vartheta _{0}} \nabla s \nabla \vartheta \\& \quad \leq M^{\ast \ast }(x, y) \biggl( \int _{\vartheta _{0}}^{x}\bigl(x- \rho (s)\bigr) \biggl( k(s)\Phi \biggl( \frac{\omega _{1}(s)}{k(s)} \biggr) \biggr) ^{2}\nabla s \biggr) ^{\frac{1}{2}} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \biggl( l(\vartheta )\Psi \biggl( \frac{\omega _{2}(\vartheta )}{l(\vartheta )} \biggr) \biggr) ^{2}\nabla \vartheta \biggr) ^{\frac{1}{2}}, \end{aligned}$$
(43)

where

$$ M^{\ast \ast }(x, y)=\frac{1}{2} \biggl( \int _{\vartheta _{0}}^{x} \biggl( \frac{\Phi (K(s))}{K(s)} \biggr) ^{2}\nabla s \biggr) ^{ \frac{1}{2}} \biggl( \int _{\vartheta _{0}}^{y} \biggl( \frac{\Psi (L(\vartheta ))}{L(\vartheta )} \biggr) ^{2}\nabla \vartheta \biggr) ^{\frac{1}{2}}, $$

which is [33, Theorem 3.5].

Remark 3.8

As a particular case of Theorem 3.6 if \(\mathbb{T}=\mathbb{Z}\), \(\vartheta _{0}=0\), then \(\rho (u)=u-1\) and (34) reduces to

$$\begin{aligned}& \sum_{s=1}^{p}\sum _{\vartheta =1}^{q} \frac{\Phi (\Pi _{s})\Psi (\Omega _{\vartheta })}{\mu s^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda \vartheta ^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \\& \quad \leq M_{0}(\lambda , \mu , p, q) \Biggl( \sum _{s=1}^{p}(p-s+1) \biggl( k_{s}\Phi \biggl( \frac{\omega _{s}}{k_{s}} \biggr) \biggr) ^{\lambda } \Biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \Biggl( \sum_{\vartheta =1}^{q}(q- \vartheta +1) \biggl( l_{ \vartheta }\Phi \biggl( \frac{\omega _{\vartheta }}{l_{\vartheta }} \biggr) \biggr) ^{\mu } \Biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(44)

where

$$ M_{0}(\lambda ,\mu ,p,q)=\frac{1}{\lambda +\mu } \Biggl( \sum _{s=1}^{p} \biggl( \frac{\Phi (K_{s})}{K_{s}} \biggr) ^{ \frac{\lambda }{\lambda -1}} \Biggr) ^{\frac{\lambda -1}{\lambda }} \Biggl( \sum _{\vartheta =1}^{q} \biggl( \frac{\Psi (L_{\vartheta })}{L_{\vartheta }} \biggr) ^{ \frac{\mu }{\mu -1}} \Biggr) ^{\frac{\mu -1}{\mu }}, $$

which is [13, Theorem 2.2].

Remark 3.9

As a particular case of Theorem 3.6 if \(\mathbb{T}=\mathbb{R}\), \(t_{0}=0\), then \(\rho (u)=u\) and (34) reduces to

$$\begin{aligned}& \int _{0}^{x} \int _{0}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))}{\mu s^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda \vartheta ^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\,ds\,dt \\& \quad \leq M_{0}^{\ast }(\lambda , \mu , x, y) \biggl( \int _{0}^{x}(x-s) \biggl( k(s)\Phi \biggl( \frac{\omega _{1}(s)}{k(s)} \biggr) \biggr) ^{\lambda }\,ds \biggr) ^{ \frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{0}^{y}(y-\vartheta ) \biggl( l( \vartheta ) \Psi \biggl( \frac{\omega _{2}(\vartheta )}{l(\vartheta )} \biggr) \biggr) ^{\mu } \,d\vartheta \biggr) ^{\frac{1}{\mu }}. \end{aligned}$$
(45)

where

$$\begin{aligned} M_{0}^{\ast }(\lambda ,\mu ,x,y) =&\frac{1}{\lambda +\mu } \biggl( \int _{0}^{x} \biggl( \frac{\Phi (K(s))}{K(s)} \biggr) ^{ \frac{\lambda }{\lambda -1}}\,ds \biggr) ^{\frac{\lambda -1}{\lambda }} \\ &{}\times \biggl( \int _{0}^{y} \biggl( \frac{\Psi (L(\vartheta ))}{L(\vartheta )} \biggr) ^{\frac{\mu }{\mu -1}}\,d\vartheta \biggr) ^{\frac{\mu -1}{\mu }}, \end{aligned}$$

which is [13, Theorem 3.2].

Our next outcome deals with a further generalization of the inequality in (34).

Theorem 3.10

Let \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\), and \(\omega _{1}\), \(\omega _{2}\) be as in Theorem 3.1. Define

$$ \Pi (s)=\frac{1}{s-\vartheta _{0}} \int _{\vartheta _{0}}^{s}\omega _{1}( \xi ) \nabla \xi \quad \textit{and}\quad \Omega (\vartheta )= \frac{1}{\vartheta -\vartheta _{0}} \int _{\vartheta _{0}}^{\vartheta } \omega _{2}(\xi ) \nabla \xi . $$
(46)

Then for \(s\in {}[ \vartheta _{0}, x]_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, y]_{\mathbb{T}}\), we have

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))(s-\vartheta _{0})(\vartheta -\vartheta _{0})}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\nabla s \nabla \vartheta \\& \quad \leq N(\lambda , \mu , x, y) \biggl( \int _{ \vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \bigl( \Phi \bigl(\omega _{1}(s)\bigr) \bigr) ^{ \lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl( \Psi \bigl(\omega _{2}(\vartheta )\bigr) \bigr) ^{\mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(47)

where

$$ N(\lambda , \mu , x, y)=\frac{1}{\lambda +\mu }(x-\vartheta _{0})^{\frac{\lambda -1}{\lambda }}(y-\vartheta _{0})^{ \frac{\mu -1}{\mu }}. $$

Proof

Based on the assumptions and the inequality of Jensen (12), we can see that

$$\begin{aligned} \Phi \bigl(\Pi (s)\bigr) =&\Phi \biggl( \frac{1}{s-\vartheta _{0}} \int _{ \vartheta _{0}}^{s}\omega _{1}(\xi ) \nabla \xi \biggr) \\ \leq &\frac{1}{s-\vartheta _{0}} \int _{\vartheta _{0}}^{s}\Phi \bigl( \omega _{1}(\xi ) \bigr) \nabla \xi . \end{aligned}$$
(48)

By applying (11) to (48) with indices λ, \(\lambda /(\lambda -1)\), we have

$$ \Phi \bigl(\Pi (s)\bigr)\leq \frac{1}{s-\vartheta _{0}}(s-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }} \biggl( \int _{\vartheta _{0}}^{s}\bigl(\Phi \bigl( \omega _{1}(\xi ) \bigr) \bigr)^{\lambda }\nabla \xi \biggr) ^{ \frac{1}{\lambda }}. $$
(49)

This implies that

$$ \Phi \bigl(\Pi (s)\bigr) (s-\vartheta _{0})\leq (s-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }} \biggl( \int _{\vartheta _{0}}^{s}\bigl(\Phi \bigl( \omega _{1}( \xi ) \bigr) \bigr)^{{\lambda }}\nabla \xi \biggr) ^{ \frac{1}{^{\lambda }}}. $$
(50)

Analogously,

$$ \Psi \bigl(\Omega (\vartheta )\bigr) (\vartheta -\vartheta _{0}) \leq (\vartheta - \vartheta _{0})^{\frac{\mu -1}{\mu }} \biggl( \int _{\vartheta _{0}}^{t}\bigl( \Psi \bigl( \omega _{2}(\xi ) \bigr) \bigr)^{\mu }\nabla \xi \biggr) ^{ \frac{1}{\mu }}. $$
(51)

From (50) and (51), we get

$$\begin{aligned}& \Phi \bigl(\Pi (s)\bigr)\Psi \bigl(\Omega (\vartheta )\bigr) (s-\vartheta _{0}) ( \vartheta -\vartheta _{0}) \\& \quad \leq (s-\vartheta _{0})^{\frac{\lambda -1}{\lambda }}(\vartheta - \vartheta _{0})^{\frac{\mu -1}{\mu }} \biggl( \int _{\vartheta _{0}}^{s}\bigl( \Phi \bigl( \omega _{1}(\xi ) \bigr) \bigr)^{{\lambda }}\nabla \xi \biggr) ^{\frac{1}{^{\lambda }}} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{t}\bigl(\Psi \bigl( \omega _{2}( \xi ) \bigr) \bigr)^{\mu }\nabla \xi \biggr) ^{\frac{1}{\mu }}. \end{aligned}$$
(52)

Applying (24) to the term \((s-\vartheta _{0})^{(\lambda -1)/\lambda }\times (\vartheta - \vartheta _{0})^{(\mu -1)/\mu }\) gives

$$\begin{aligned}& \Phi \bigl(\Pi (s)\bigr)\Psi \bigl(\Omega (\vartheta )\bigr) (s-\vartheta _{0}) ( \vartheta -\vartheta _{0}) \\& \quad \leq \frac{\lambda \mu }{\lambda +\mu } \biggl( \frac{(s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}{\lambda }+ \frac{(\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\mu } \biggr) \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{s}\bigl(\Phi \bigl( \omega _{1}( \xi ) \bigr) \bigr)^{{\lambda }}\nabla \xi \biggr) ^{ \frac{1}{^{\lambda }}} \biggl( \int _{\vartheta _{0}}^{t}\bigl(\Psi \bigl( \omega _{2}(\xi ) \bigr) \bigr)^{\mu }\nabla \xi \biggr) ^{ \frac{1}{\mu }}. \end{aligned}$$
(53)

From (53), we have

$$\begin{aligned}& \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))(s-\vartheta _{0})(\vartheta -\vartheta _{0})}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \\& \quad \leq \frac{1}{\lambda +\mu } \biggl( \int _{\vartheta _{0}}^{s}\bigl( \Phi \bigl( \omega _{1}(\xi ) \bigr) \bigr)^{{\lambda }}\nabla \xi \biggr) ^{\frac{1}{^{\lambda }}} \biggl( \int _{\vartheta _{0}}^{t}\bigl(\Psi \bigl( \omega _{2}( \xi ) \bigr) \bigr)^{\mu }\nabla \xi \biggr) ^{\frac{1}{\mu }}. \end{aligned}$$
(54)

Integrating both sides of (54) and using (11) again with indices λ, \(\lambda /(\lambda -1)\) and μ, \(\mu /(\mu -1)\), we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))(s-\vartheta _{0})(\vartheta -\vartheta _{0})}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\nabla s \nabla \vartheta \\& \quad \leq \frac{1}{\lambda +\mu }(x-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }}(y- \vartheta _{0})^{\frac{\mu -1}{\mu }} \biggl( \int _{\vartheta _{0}}^{x} \biggl( \int _{\vartheta _{0}}^{s}\bigl(\Phi \bigl( \omega _{1}(\xi ) \bigr) \bigr)^{\lambda }\nabla \xi \biggr) \nabla s \biggr) ^{ \frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y} \biggl( \int _{\vartheta _{0}}^{ \vartheta }\bigl(\Psi \bigl( \omega _{2}(\xi ) \bigr) \bigr)^{\mu }\nabla \xi \biggr) \nabla \vartheta \biggr) ^{\frac{1}{\mu }}. \end{aligned}$$
(55)

Applying Lemma 2.5 to (55), we find that

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))(s-\vartheta _{0})(\vartheta -\vartheta _{0})}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\nabla s \nabla \vartheta \\& \quad \leq \frac{1}{\lambda +\mu }(x-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }}(y- \vartheta _{0})^{\frac{\mu -1}{\mu }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{x}\bigl(x- \rho (s)\bigr) \bigl( \Phi \bigl( \omega _{1}(s) \bigr) \bigr)^{\lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl( \Psi \bigl( \omega _{2}(\vartheta ) \bigr) \bigr)^{\mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }} \\& \quad =N(\lambda , \mu , x, y) \biggl( \int _{\vartheta _{0}}^{x}\bigl(x- \rho (s)\bigr) \bigl( \Phi \bigl( \omega _{1}(s) \bigr) \bigr)^{\lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl( \Psi \bigl( \omega _{2}(\vartheta ) \bigr) \bigr)^{\mu }\nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$

which is (47). □

Corollary 3.3

If we take \(1/\lambda +1/\mu =1\) in (47), then we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))(s-\vartheta _{0})(\vartheta -\vartheta _{0})}{\mu (s-\vartheta _{0})^{{\lambda -1}}+\lambda (\vartheta -\vartheta _{0})^{{\mu -1}}} \nabla s \nabla \vartheta \\ & \quad \leq N^{\ast }(\lambda , \mu , x, y) \biggl( \int _{ \vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \bigl(\Phi \bigl( \omega _{1}(s) \bigr) \bigr)^{ \lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\ & \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl(\Psi \bigl( \omega _{2}(\vartheta ) \bigr) \bigr)^{\mu } \nabla \vartheta \biggr) ^{ \frac{1}{\mu }} , \end{aligned}$$
(56)

where

$$ N^{\ast }(\lambda , \mu , x, y)= \frac{1}{\lambda \mu }(x- \vartheta _{0})^{\frac{\lambda -1}{\lambda }}(y-\vartheta _{0})^{ \frac{\mu -1}{\mu }}. $$

Remark 3.11

As a particular case of Corollary 3.3, if \(\lambda =\mu =2\), then we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))(s-\vartheta _{0})(\vartheta -\vartheta _{0})}{s+\vartheta -2\vartheta _{0}}\nabla s \nabla \vartheta \\ & \quad \leq \frac{1}{2} \biggl( (x-\vartheta _{0}) \int _{\vartheta _{0}}^{x}\bigl(x- \rho (s)\bigr) \bigl( \Phi \bigl(\omega _{1}(s)\bigr) \bigr) ^{2}\nabla s \biggr) ^{ \frac{1}{2}} \\ & \qquad {}\times \biggl( (y-\vartheta _{0}) \int _{\vartheta _{0}}^{y}\bigl(y-\rho ( \vartheta )\bigr) \bigl( \Psi \bigl(\omega _{2}(\vartheta )\bigr) \bigr) ^{2} \nabla \vartheta \biggr) ^{\frac{1}{2}}, \end{aligned}$$
(57)

which is [33, Theorem 3.6].

Remark 3.12

As a particular case of Theorem 3.10, if \(\mathbb{T}=\mathbb{Z}\), \(\vartheta _{0}=0\), then \(\rho (u)=u-1\) and (47) reduces to

$$\begin{aligned}& \sum_{s=1}^{p}\sum _{\vartheta =1}^{q} \frac{\Phi (\Pi _{s})\Psi (\Omega _{\vartheta })s\vartheta }{\mu s^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda \vartheta ^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \\ & \quad \leq N_{0}(\lambda , \mu , p, q) \Biggl( \sum _{s=1}^{p}(p-s+1) \bigl( \Phi ( \omega _{s} ) \bigr) ^{\lambda } \Biggr) ^{ \frac{1}{\lambda }} \Biggl( \sum_{\vartheta =1}^{q}(q- \vartheta +1) \bigl( \Phi ( \omega _{\vartheta } ) \bigr) ^{\mu } \Biggr) ^{ \frac{1}{\mu }}, \end{aligned}$$
(58)

where

$$ N_{0}(\lambda , \mu , p, q)=\frac{1}{\lambda +\mu }p^{\frac{\lambda -1}{\lambda }}q^{\frac{\mu -1}{\mu }}, $$

which is [13, Theorem 2.3].

Remark 3.13

As a particular state of Theorem 3.10, if \(\mathbb{T}=\mathbb{R}\), \(t_{0}=0\), then \(\rho (u)=u\) and (47) reduces to

$$\begin{aligned}& \int _{0}^{x} \int _{0}^{y} \frac{\Phi (\Pi (s))\Psi (\Omega (\vartheta ))s\vartheta }{\mu s^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda \vartheta ^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\,ds\,d\vartheta \\ & \quad \leq N_{0}^{\ast }(\lambda , \mu , x, y) \biggl( \int _{0}^{x}(x-s) \bigl(\Phi \bigl( \omega _{1}(s) \bigr) \bigr)^{\lambda }\,ds \biggr) ^{\frac{1}{^{\lambda }}} \\& \qquad {}\times \biggl( \int _{0}^{y}(y-\vartheta ) \bigl(\Psi \bigl( \omega _{2}( \vartheta ) \bigr) \bigr)^{\mu }\,d\vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(59)

where

$$ N_{0}^{\ast }(\lambda , \mu , x, y)= \frac{1}{\lambda +\mu }x^{\frac{\lambda -1}{\lambda }}y^{ \frac{\mu -1}{\mu }}, $$

which is [13, Theorem 3.3].

Theorem 3.14

Let \(s, \vartheta , \vartheta _{0}\in \mathbb{T}\) and \(\omega _{1}\), \(\omega _{2}\), k, l, H, L be as in Theorem 3.6. Define

$$ \Pi (s)=\frac{1}{K(s)} \int _{\vartheta _{0}}^{s}k(\xi )\omega _{1}( \xi )\nabla \xi \quad \textit{and}\quad \Omega (\vartheta )= \frac{1}{L(\vartheta )}\int _{\vartheta _{0}}^{\vartheta }l(\xi )\omega _{2}(\xi )\nabla \xi . $$
(60)

Then for \(s\in {}[ \vartheta _{0}, y]_{\mathbb{T}}\) and \(\vartheta \in {}[ \vartheta _{0}, x]_{\mathbb{T}}\), we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{K(s)L(\vartheta )\Phi (\Pi (s))\Psi (\Omega (\vartheta ))}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq W(\lambda , \mu , x, y) \biggl( \int _{ \vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \bigl(k(s) \Phi \bigl( \omega _{1}(s) \bigr) \bigr)^{\lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl(l( \vartheta )\Psi \bigl( \omega _{2}(\vartheta ) \bigr) \bigr)^{\mu } \nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(61)

where

$$ W(\lambda , \mu , x, y)=\frac{1}{\lambda +\mu }(x-\vartheta _{0})^{\frac{\lambda -1}{\lambda }}(y-\vartheta _{0})^{ \frac{\mu -1}{\mu }}. $$

Proof

Based on the assumptions and the inequality of Jensen (12), we find that

$$\begin{aligned} \Phi \bigl(\Pi (s)\bigr) =&\Phi \biggl( \frac{1}{K(s)} \int _{\vartheta _{0}}^{s}k( \xi )\omega _{1}( \xi )\nabla \xi \biggr) \\ \leq &\frac{1}{K(s)} \int _{\vartheta _{0}}^{s}k(\xi )\Phi \bigl(\omega _{1}( \xi )\bigr)\nabla \xi . \end{aligned}$$
(62)

By applying (11) to (62) with indices λ, \(\lambda /(\lambda -1)\), we have

$$ \Phi \bigl(\Pi (s)\bigr)\leq \frac{1}{K(s)}(s-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }} \biggl( \int _{\vartheta _{0}}^{s} \bigl( k(\xi )\Phi \bigl( \omega _{1}(\xi )\bigr) \bigr) ^{\lambda }\nabla \xi \biggr) ^{ \frac{1}{\lambda }}. $$
(63)

From (63), we get

$$ \Phi \bigl(\Pi (s)\bigr)K(s)\leq (s-\vartheta _{0})^{{ \frac{\lambda -1}{\lambda }}} \biggl( \int _{\vartheta _{0}}^{s} \bigl( k(\xi )\Phi \bigl(\omega _{1}( \xi )\bigr) \bigr) ^{\lambda }\nabla \xi \biggr) ^{\frac{1}{\lambda }}. $$
(64)

Similarly, we also obtain

$$ \Psi (\Omega (\vartheta )L(\vartheta )\leq (\vartheta -\vartheta _{0})^{\frac{\mu -1}{\mu }} \biggl( \int _{\vartheta _{0}}^{\vartheta } \bigl( l(\xi )\Psi \bigl(\omega _{2}(\xi )\bigr) \bigr) ^{\mu }\nabla \xi \biggr) ^{ \frac{1}{\mu }}. $$
(65)

From (64) and (65), we find that

$$\begin{aligned}& K(s)L(\vartheta )\Phi \bigl(\Pi (s)\bigr)\Psi (\Omega (\vartheta ) \\& \quad \leq (s-\vartheta _{0})^{{\frac{\lambda -1}{\lambda }}}(\vartheta - \vartheta _{0})^{\frac{\mu -1}{\mu }} \biggl( \int _{\vartheta _{0}}^{s} \bigl( k(\xi )\Phi \bigl(\omega _{1}(\xi )\bigr) \bigr) ^{\lambda }\nabla \xi \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{\vartheta } \bigl( l(\xi ) \Psi \bigl(\omega _{2}(\xi )\bigr) \bigr) ^{\mu }\nabla \xi \biggr) ^{ \frac{1}{\mu }}. \end{aligned}$$
(66)

Applying (24) to the term \((s-\vartheta _{0})^{(\lambda -1)/\lambda }\times (\vartheta - \vartheta _{0})^{(\mu -1)/\mu }\) gives

$$\begin{aligned}& K(s)L(\vartheta )\Phi \bigl(\Pi (s)\bigr)\Psi (\Omega (\vartheta ) \\& \quad \leq \frac{\lambda \mu }{\lambda +\mu } \biggl( \frac{(s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}}{\lambda }+ \frac{(\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}{\mu } \biggr) \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{s} \bigl( k(\xi )\Phi \bigl(\omega _{1}( \xi )\bigr) \bigr) ^{\lambda }\nabla \xi \biggr) ^{\frac{1}{\lambda }} \biggl( \int _{\vartheta _{0}}^{\vartheta } \bigl( l(\xi )\Psi \bigl( \omega _{2}(\xi )\bigr) \bigr) ^{\mu }\nabla \xi \biggr) ^{ \frac{1}{\mu }}. \end{aligned}$$
(67)

This implies that

$$\begin{aligned}& \frac{K(s)L(\vartheta )\Phi (\Pi (s))\Psi (\Omega (\vartheta )}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \\& \quad \leq \frac{1}{\lambda +\mu } \biggl( \int _{\vartheta _{0}}^{s} \bigl( k(\xi )\Phi \bigl(\omega _{1}(\xi )\bigr) \bigr) ^{\lambda }\nabla \xi \biggr) ^{\frac{1}{\lambda }} \biggl( \int _{\vartheta _{0}}^{\vartheta } \bigl( l(\xi ) \Psi \bigl(\omega _{2}(\xi )\bigr) \bigr) ^{\mu }\nabla \xi \biggr) ^{ \frac{1}{\mu }}. \end{aligned}$$
(68)

Integrating both sides of (68) and using (11) again with indices λ, \(\lambda /(\lambda -1)\) and μ, \(\mu /(\mu -1)\), we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{K(s)L(\vartheta )\Phi (\Pi (s))\Psi (\Omega (\vartheta )}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq \frac{1}{\lambda +\mu }(s-\vartheta _{0})^{ \frac{\lambda -1}{\lambda }}( \vartheta -\vartheta _{0})^{\frac{\mu -1}{\mu }} \biggl( \int _{ \vartheta _{0}}^{x} \biggl( \int _{\vartheta _{0}}^{s} \bigl( k(\xi ) \Phi \bigl(\omega _{1}(\xi )\bigr) \bigr) ^{\lambda }\nabla \xi \biggr) \nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y} \biggl( \int _{\vartheta _{0}}^{ \vartheta } \bigl( l(\xi )\Psi \bigl(\omega _{2}(\xi )\bigr) \bigr) ^{\mu } \nabla \xi \biggr) \nabla \vartheta \biggr) ^{\frac{1}{\mu }}. \end{aligned}$$
(69)

Applying Lemma 2.5 to (69), we find that

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{K(s)L(\vartheta )\Phi (\Pi (s))\Psi (\Omega (\vartheta )}{\mu (s-\vartheta _{0})^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda (\vartheta -\vartheta _{0})^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \nabla s \nabla \vartheta \\& \quad \leq W(\lambda , \mu , x, y) \biggl( \int _{ \vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \bigl(k(s) \Phi \bigl( \omega _{1}(s) \bigr) \bigr)^{\lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\& \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl(l( \vartheta )\Psi \bigl( \omega _{2}(\vartheta ) \bigr) \bigr)^{\mu } \nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$

which is (61). □

Corollary 3.4

If we take \(1/\lambda +1/\mu =1\) in (61), then

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{K(s)L(\vartheta )\Phi (\Pi (s))\Psi (\Omega (\vartheta )}{\mu (s-\vartheta _{0})^{{\lambda -1}}+\lambda (\vartheta -\vartheta _{0})^{{\mu -1}}} \nabla s \nabla \vartheta \\ & \quad \leq W^{\ast }(\lambda , \mu , x, y) \biggl( \int _{ \vartheta _{0}}^{x}\bigl(x-\rho (s)\bigr) \bigl(k(s) \Phi \bigl( \omega _{1}(s) \bigr) \bigr)^{\lambda }\nabla s \biggr) ^{\frac{1}{\lambda }} \\ & \qquad {}\times \biggl( \int _{\vartheta _{0}}^{y}\bigl(y-\rho (\vartheta )\bigr) \bigl(l( \vartheta )\Psi \bigl( \omega _{2}(\vartheta ) \bigr) \bigr)^{\mu } \nabla \vartheta \biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(70)

where

$$ W^{\ast }(\lambda , \mu , x, y)= \frac{1}{\lambda \mu }(x- \vartheta _{0})^{\frac{\lambda -1}{\lambda }}(y-\vartheta _{0})^{ \frac{\mu -1}{\mu }}. $$

Remark 3.15

As a particular case of Corollary 3.4, if \(\lambda =\mu =2\), then we get

$$\begin{aligned}& \int _{\vartheta _{0}}^{x} \int _{\vartheta _{0}}^{y} \frac{K(s)L(\vartheta )\Phi (\Pi (s))\Psi (\Omega (\vartheta )}{s+\vartheta -2\vartheta _{0}}\nabla s\nabla \vartheta \\ & \quad \leq \frac{1}{2} \biggl( (x-\vartheta _{0}) \int _{\vartheta _{0}}^{x}\bigl(x- \rho (s)\bigr) \bigl( k(s)\Phi \bigl(\omega _{1}(s)\bigr) \bigr) ^{2}\nabla s \biggr) ^{\frac{1}{2}} \\ & \qquad {}\times \biggl( (y-\vartheta _{0}) \int _{\vartheta _{0}}^{y}\bigl(y-\rho ( \vartheta )\bigr) \bigl( l(\vartheta )\Psi \bigl(\omega _{2}(\vartheta )\bigr) \bigr) ^{2}\nabla \vartheta \biggr) ^{\frac{1}{2}}, \end{aligned}$$
(71)

which is [33, Theorem 3.7].

Remark 3.16

As a particular case of Theorem 3.14, if \(\mathbb{T}=\mathbb{Z}\), \(\vartheta _{0}=0\), then \(\rho (u)=u-1\) and (61) reduces to

$$\begin{aligned} \sum_{s=1}^{p}\sum _{\vartheta =1}^{q} \frac{K_{s}L_{\vartheta }\Phi (\Pi _{s})\Psi (\Omega _{\vartheta })}{\mu s^{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}+\lambda \vartheta ^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}} \leq& W_{0}(\lambda , \mu , p, q) \Biggl( \sum _{s=1}^{p}(p-s+1) \bigl( k_{s}\Phi ( \omega _{s} ) \bigr) ^{\lambda } \Biggr) ^{\frac{1}{\lambda }} \\ & {}\times \Biggl( \sum_{\vartheta =1}^{q}(q- \vartheta +1) \bigl( l_{ \vartheta }\Phi ( \omega _{\vartheta } ) \bigr) ^{\mu } \Biggr) ^{\frac{1}{\mu }}, \end{aligned}$$
(72)

where

$$ W_{0}(\lambda , \mu , p, q)=\frac{1}{\lambda +\mu }p^{\frac{\lambda -1}{\lambda }}q^{\frac{\mu -1}{\mu }}, $$

which is [13, Theorem 2.4].

Remark 3.17

As a particular case of Theorem 3.14, if \(\mathbb{T}=\mathbb{R}\), \(t_{0}=0\), then \(\rho (u)=u\) and (61) reduces to

$$\begin{aligned}& \int _{0}^{x} \int _{0}^{y} \frac{K(s)L(\vartheta )\Phi (\Pi (s))\Psi (\Omega (\vartheta ))}{\mu s^{{\frac{(\lambda -1)(\lambda +\mu )}{\lambda \mu }}}+\lambda \vartheta ^{{\frac{(\mu -1)(\lambda +\mu )}{\lambda \mu }}}}\,ds\,d\vartheta \\ & \quad \leq W_{0}^{\ast }(\lambda , \mu , x, y) \biggl( \int _{0}^{x}(x-s) \bigl(k(s)\Phi \bigl( \omega _{1}(s) \bigr) \bigr)^{ \lambda }\,ds \biggr) ^{\frac{1}{^{\lambda }}} \\& \qquad {}\times \biggl( \int _{0}^{y}(y-\vartheta ) \bigl(l(\vartheta )\Psi \bigl( \omega _{2}(\vartheta ) \bigr) \bigr)^{\mu } \,d\vartheta \biggr) ^{ \frac{1}{\mu }}, \end{aligned}$$
(73)

where

$$ W_{0}^{\ast }(\lambda , \mu , x, y)= \frac{1}{\lambda +\mu }x^{\frac{\lambda -1}{\lambda }}y^{ \frac{\mu -1}{\mu }}, $$

which is [13, Theorem 3.4].

Remark 3.18

Clearly, Theorems 3.1, 3.6, 3.10, and 3.14 present the corresponding results of Theorems 6, 9, 12, and 15 in [18], respectively, for time scale delta calculus. Likewise, Corollaries 3.1, 3.2, 3.3, and 3.4 display the corresponding results of Theorems 3.1, 3.2, 3.3, and 3.4 in [17], respectively, for delta time scale calculus.