Some New Inverse Hilbert Inequalities on Time Scales

Abstract

Several inverse integral inequalities were proved in 2004 by Yong. It is our aim in this paper to extend these inequalities to time scales. Furthermore, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Our results are proved using some algebraic inequalities, inverse Hölder’s inequality and inverse Jensen’s inequality on time scales. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

1. Introduction

The form of the established classical discrete Hardy–Hilbert double series inequality [1] is given as follows: If $\left\{a_m\right\}⩾0$, $\left\{b_n\right\}⩾0$, $0<{\sum }_{n=1}^{\infty }{a}_n^p<\infty$ and $0<{\sum }_{n=1}^{\infty }{b}_n^q<\infty ,$ then we have

$$\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\displaystyle \frac{a_n{b}_m}{m+n}}⩽\frac{\pi }{sin\frac{\pi }{p}}{\left(\sum _{n=1}^{\infty }{a}_n^p\right)}^{\frac{1}{p}}{\left(\sum _{m=1}^{\infty }{b}_m^q\right)}^{\frac{1}{q}},$$

(1)

where $p>1$, $q=\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$p-1$}\right.$.

The continuous versions of inequality (1) is given by:

$${\int }_0^{\infty }{\int }_0^{\infty }\frac{f\left(x\right)g\left(y\right)}{x+y}dxdy⩽\frac{\pi }{sin\frac{\pi }{p}}{\left({\int }_0^{\infty }{f}^p\left(x\right)dx\right)}^{\frac{1}{p}}{\left({\int }_0^{\infty }{g}^{p^{\prime }}\left(x\right)dx\right)}^{\frac{1}{q}},$$

(2)

unless $f\equiv 0$ or $g\equiv 0,$ where f and g are measurable non-negative functions such that ${\int }_0^{\infty }{f}^p\left(x\right)dx<\infty$ and ${\int }_0^{\infty }{g}^p\left(x\right)dx<\infty$. The constant $\frac{\pi }{sin\frac{\pi }{p}},$ in (1) and (2), is the best possible.

In [2], Pachpatte proved that if $f\in C^1[[0,x],{\mathbb{R}}^{+}],$ $g\in C^1[[0,y],{\mathbb{R}}^{+}]$ with $f\left(0\right)=g\left(0\right)=0$ and $p,$q are two positive functions defined for $t\in [0,x)$ and $\tau \in [0,y),$ with $P\left(t\right)={\int }_0^{t}p\left(\tau \right)d\tau$ and $Q\left(t\right)={\int }_0^{t}q\left(\tau \right)d\tau$ for $s\in [0,x)$ and $t\in [0,y)$, where $x,$y are positive real numbers. Let $\Phi$ and $\Psi$ be two real-valued non-negative, convex and sub-multiplicative functions defined on $[0,\infty ).$ Then,

$$\begin{array}{cc}{\int }_0^x{\int }_0^y{\displaystyle \frac{\Phi \left(f\right(s\left)\right)\Psi \left(g\right(t\left)\right)}{s+t}}dsdt⩽\hfill & L(x,y)({\int }_0^x(x-s){\left(p\left(s\right)\Phi {\left(\frac{f^{\prime }\left(s\right)}{p\left(s\right)}\right)}^{2}ds\right)}^{\frac{1}{2}}\hfill \\ \hfill & \times ({\int }_0^y(y-t){\left(q\left(t\right)\Psi {\left(\frac{g^{\prime }\left(t\right)}{q\left(t\right)}\right)}^{2}dt\right)}^{\frac{1}{2}},\hfill \end{array}$$

(3)

where

$$L(x,y)=\frac{1}{2}{\left({\int }_0^x{\left(\frac{\Phi \left(P\right(s\left)\right)}{P\left(s\right)}\right)}^{2}ds\right)}^{\frac{1}{2}}{\left({\int }_0^y{\left(\frac{\Psi \left(Q\right(t\left)\right)}{Q\left(t\right)}\right)}^{2}dt\right)}^{\frac{1}{2}}.$$

In 2004, Yong [3] studied the following integral inequality:

Theorem 1.

Let $l,$$m⩾1$$r⩽0$ and $f\left(\sigma \right)⩾0,$ $g\left(\tau \right)⩾0$ for $\sigma \in (0,\xi ),$ $\tau \in (0,\zeta )$, where $\xi ,$ζ are positive real numbers and define

$$\Theta (\Im ):={\int }_0^{\Im }f\left(\sigma \right)d\sigma ,and\Xi \left(\varsigma \right):={\int }_0^{\varsigma }g\left(\tau \right)d\tau ,$$

for $\Im \in (0,\xi )$ and $\varsigma \in (0,\zeta ).$ Then, for $p^{-1}+q^{-1}=1,$ $p<0$ or $0<p<1$

$$\begin{array}{ccc}\hfill {\int }_0^{\xi }{\int }_0^{\zeta }\frac{{\Theta }^l(\Im ){\Xi }^m\left(\varsigma \right)}{{\left(\frac{({\Im }^r+{\varsigma }^r)}{2}\right)}^{\frac{2}{rp}}}d\Im d\varsigma & ⩾& lm{\left(\xi \zeta \right)}^{\frac{1}{p}}{\left({\int }_0^{\xi }(\xi -\Im ){\left({\Theta }_f(\Im )\right)}^{q}d\Im \right)}^{\frac{1}{q}}\hfill \\ & & \times {\left({\int }_0^{\zeta }(\zeta -\varsigma ){\left({\Xi }_g\left(\varsigma \right)\right)}^{q}d\varsigma \right)}^{\frac{1}{q}}\hfill \end{array}$$

(4)

unless $f\equiv 0$ or $g\equiv 0,$ where ${\Theta }_f(\Im )={\Theta }^{l-1}(\Im )f(\Im ),$ ${\Xi }_g\left(\varsigma \right)={\Xi }^{m-1}\left(\varsigma \right)g\left(\varsigma \right).$

In 2009, Yang [4] studied the following integral inequality:

Theorem 2.

Let $p,$ $q⩾0,$ $\alpha >1,$ $\gamma >1$ and $f\left(\sigma \right)⩾0,$ $g\left(\tau \right)⩾0$ for $\sigma \in (0,\xi ),$ $\tau \in (0,\zeta )$, where $\xi ,$ ζ are positive real numbers and define

$$\Theta (\Im ):={\int }_0^{\Im }f\left(\sigma \right)d\sigma ,and\Xi \left(\varsigma \right):={\int }_0^{\varsigma }g\left(\tau \right)d\tau ,$$

for $\Im \in (0,\xi )$ and $\varsigma \in (0,\zeta ).$ Then,

$$\begin{array}{ccc}\hfill {\int }_0^{\xi }{\int }_0^{\zeta }\frac{{\Theta }^p(\Im ){\Xi }^q\left(\varsigma \right)}{\gamma {\Im }^{\frac{(\alpha -1)(\alpha +\gamma )}{\alpha \gamma }}+\alpha {\varsigma }^{\frac{(\alpha -1)(\alpha +\gamma )}{\alpha \gamma }}}d\Im d\varsigma & ⩽& D(p,q,\xi ,\zeta ,\alpha ,\gamma ){\left({\int }_0^{\xi }(\xi -\Im ){\left({\Theta }^{p-1}(\Im )f(\Im )\right)}^{\alpha }d\Im \right)}^{\frac{1}{\alpha }}\hfill \\ & & \times {\left({\int }_0^{\zeta }(\zeta -\varsigma ){\left({\Xi }^{q-1}\left(\varsigma \right)g\left(\varsigma \right)\right)}^{\gamma }d\varsigma \right)}^{\frac{1}{\gamma }}\hfill \end{array}$$

(5)

unless $f\equiv 0$ or $g\equiv 0,$ where $D(p,q,\xi ,\zeta ,\alpha ,\gamma )=\frac{pq}{\alpha +\gamma }{\xi }^{\frac{\alpha -1}{\alpha }}{\zeta }^{\frac{\gamma -1}{\gamma }}.$

In this paper, we prove some new dynamic inequalities of Hilbert type and their converses on time scales. From these inequalities, as special cases, we formulate some special integral and discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

Now, we present some fundamental concepts and effects on time scales which are beneficial for deducing our main results. In 1988, S. Hilger [5] presented time scale theory to unify continuous and discrete analysis. For some Hilbert-type integral, dynamic inequalities and other types of inequalities on time scales, see the papers [2,3,6,7,8,9,10,11,12,13,14,15,16]. For more details on time scale calculus see [17].

We need the following important relations between calculus on time scales $\mathbb{T}$ and either continuous calculus on $\mathbb{R}$ or discrete calculus on $\mathbb{Z}$. Note that:

(i)

If $\mathbb{T}=\mathbb{R}$, then

$$\sigma \left(\varsigma \right)=\varsigma ,\mu \left(\varsigma \right)=0,f^{\Delta }\left(\varsigma \right)=f^{\prime }\left(\varsigma \right),{\int }_a^{b}f\left(\varsigma \right)\Delta \varsigma ={\int }_a^{b}f\left(\varsigma \right)d\varsigma .$$

(6)

($ii$)

If $\mathbb{T}=\mathbb{Z}$, then

$$\sigma \left(\varsigma \right)=\varsigma +1,\mu \left(\varsigma \right)=1,f^{\Delta }\left(\varsigma \right)=f(\varsigma +1)-f\left(\varsigma \right),{\int }_a^{b}f\left(\varsigma \right)\Delta \varsigma =\sum _{\varsigma =a}^{b-1}f\left(\varsigma \right).$$

(7)

Next, we write Hölder’s inequality and Jensen’s inequality on time scales.

Lemma 1

(Dynamic Hölder’s Inequality [18]). Let $a,b\in \mathbb{T}$ and f, $g\in C_{rd}({[a,b]}_{\mathbb{T}},[0,\infty ))$. If p, $q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, then

$${\int }_a^{b}f\left(\varsigma \right)g\left(\varsigma \right)\Delta \varsigma \le {\left[{\int }_a^b{f}^p\left(\varsigma \right)\Delta \varsigma \right]}^{\frac{1}{p}}{\left[{\int }_a^b{g}^q\left(\varsigma \right)\Delta \varsigma \right]}^{\frac{1}{q}}.$$

(8)

This inequality is reversed if $0<p<1$ and if $p<0$ or $q<0.$

Lemma 2

(Dynamic Jensen’s Inequality [18]). Let a, $b\in \mathbb{T}$ and c, $d\in \mathbb{R}$. Assume that $g\in C_{rd}\left({[a,b]}_{\mathbb{T}},[c,d]\right)$ and $r\in C_{rd}\left({[a,b]}_{\mathbb{T}},\mathbb{R}\right)$ are non-negative with ${\int }_a^{b}r\left(\varsigma \right)\Delta \varsigma >0$. If $\varphi \in C_{rd}\left((c,d),\mathbb{R}\right)$ is a convex function, then

$$\varphi \left({\displaystyle \frac{{\int }_a^{b}g\left(\varsigma \right)r\left(\varsigma \right)\Delta \varsigma }{{\int }_a^{b}r\left(\varsigma \right)\Delta \varsigma }}\right)⩽{\displaystyle \frac{{\int }_a^{b}r\left(\varsigma \right)\varphi \left(g\left(\varsigma \right)\right)\Delta \varsigma }{{\int }_a^{b}r\left(\varsigma \right)\Delta \varsigma }}.$$

(9)

This inequality is reversed if $\varphi \in C_{rd}\left((c,d),\mathbb{R}\right)$ is concave.

Moreover, we use the following definition and lemma as we see in the proof of our results:

Definition 1.

Λ is called a supermultiplicative function on $[0,\infty )$ if

$$\Lambda \left(xy\right)⩾\Lambda \left(\xi \right)\Lambda \left(\zeta \right),\mathit{for}\mathit{all}\xi ,\zeta ⩾0.$$

(10)

Lemma 3

([19]). Let $\mathbb{T}$ be a time scale with $\xi ,a\in \mathbb{T}$ such that $\xi ⩾a.$ If $f⩾0$ and $\alpha ⩾1,$ then

$${\left({\int }_a^{\xi }f\left(\tau \right)\Delta \tau \right)}^{\alpha }⩾\alpha {\int }_a^{\xi }f\left(\eta \right){\left({\int }_a^{\eta }f\left(\tau \right)\Delta \tau \right)}^{\alpha -1}\Delta \eta .$$

(11)

Now, we present the formula that reduces double integrals to single integrals, which is desired in [9].

Lemma 4.

Let $\chi :\mathbb{T}⟶\mathbb{R}$ and $u,$$\Im ,$$\theta \in \mathbb{T}.$ Then,

$${\int }_u^{\Im }{\int }_u^{\theta }\chi \left(\tau \right)\Delta \tau \Delta \theta ={\int }_u^{\Im }(\Im -\sigma \left(\theta \right))\chi \left(\theta \right)\Delta \theta for\Im \in \mathbb{T},$$

(12)

assuming the integrals considered exist.

The following section contains our main results:

2. Main Results

In the next theorems, we assume that $p<0$ and $\frac{1}{p}+\frac{1}{q}=1.$

Theorem 3.

Let $\mathbb{T}$ be time scale with $L,$$K⩾1$ and $\Im ,$$\varsigma ,$${\varsigma }_0,$$\xi ,$ζ$\in \mathbb{T}.$ Assume $a\left(\tau \right)$ and $b\left(\tau \right)$ are two non-negative and right-dense continuous functions on $[{\varsigma }_0,\xi ]$ and $[{\varsigma }_0,\zeta ]$, respectively, and define

$$\psi (\Im ):={\int }_{{\varsigma }_0}^{\Im }a\left(\tau \right)\Delta \tau and\phi \left(\varsigma \right):={\int }_{{\varsigma }_0}^{\varsigma }b\left(\tau \right)\Delta \tau ,$$

then, for $\Im \in [{\varsigma }_0,\xi ]$ and $\varsigma \in [{\varsigma }_0,\zeta ],$ we have that

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }\frac{{\psi }^K(\Im ){\phi }^L\left(\varsigma \right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾C_1(K,L,p,p_{*}){\left({\int }_{{\varsigma }_0}^{\xi }(\rho \left(\xi \right)-\sigma (\Im )){\left(a(\Im ){\psi }^{K-1}(\Im )\right)}^p\Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\zeta }(\rho \left(\zeta \right)-\sigma \left(\varsigma \right)){\left(b\left(\varsigma \right){\phi }^{L-1}\left(\varsigma \right)\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}},\hfill \end{array}$$

(13)

where

$$C_1(K,L,p,p_{*})=\frac{KL}{{(p+p_{*})}^{\frac{p+p_{*}}{p{p}_{*}}}}{(\xi -{\varsigma }_0)}^{\frac{p-1}{p}}{(\zeta -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}.$$

Proof. 

By using the inequality (11), we obtain

$${\psi }^K(\Im )⩾K{\int }_{{\varsigma }_0}^{\Im }a\left(\eta \right){\psi }^{K-1}\left(\eta \right)\Delta \eta ,$$

(14)

$${\phi }^L\left(\varsigma \right)⩾L{\int }_{{\varsigma }_0}^{\varsigma }b\left(\eta \right){\phi }^{L-1}\left(\eta \right)\Delta \eta .$$

(15)

Applying inverse Hölder’s inequality on the right hand side of (14) with indices p and $\frac{p}{p-1},$ we have

$${\psi }^K(\Im )⩾K{(\Im -{\varsigma }_0)}^{\frac{p-1}{p}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left(a\left(\eta \right){\psi }^{K-1}\left(\eta \right)\right)}^p\Delta \eta \right)}^{\frac{1}{p}}.$$

(16)

Applying inverse Hölder’s inequality on the right hand side of (15) with indices $p_{*}$ and $\frac{p_{*}}{p_{*}-1},$ we also have that

$${\phi }^L\left(\varsigma \right)⩾L{(\varsigma -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(b\left(\eta \right){\phi }^{L-1}\left(\eta \right)\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}.$$

(17)

From (16) and (17), we obtain

$$\begin{array}{c}\hfill {\psi }^K(\Im ){\phi }^L\left(\varsigma \right)⩾KL{(\Im -{\varsigma }_0)}^{{}^{\frac{p-1}{p}}}{(\varsigma -{\varsigma }_0)}^{{}^{\frac{p_{*}-1}{p_{*}}}}\\ \hfill \times {\left({\int }_{{\varsigma }_0}^{\Im }{\left(a\left(\eta \right){\psi }^{K-1}\left(\eta \right)\right)}^p\Delta \eta \right)}^{\frac{1}{p}}\\ \hfill \times {\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(b\left(\eta \right){\phi }^{L-1}\left(\eta \right)\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}.\end{array}$$

(18)

Using the following inequality

$${\lambda }_1^{{\alpha }_1^{\prime }}{\lambda }_2^{{\alpha }_2^{\prime }}⩾{\left(\frac{1}{{\alpha }^{\prime }}\left({\alpha }_1^{\prime }{\lambda }_1+{\alpha }_2^{\prime }{\lambda }_2\right)\right)}^{{\alpha }^{\prime }},$$

(19)

where ${\alpha }_1^{\prime },$${\alpha }_2^{\prime }<0$ and ${\lambda }_1,$${\lambda }_2>0.$ Now, by setting ${\lambda }_1={(\Im -{\varsigma }_0)}^{(p-1)},$ ${\lambda }_2={(\varsigma -{\varsigma }_0)}^{(p_{*}-1)},$ ${\alpha }_1^{\prime }=\frac{1}{p},$ ${\alpha }_2^{\prime }=\frac{1}{p_{*}}$ and ${\alpha }^{\prime }={\alpha }_1^{\prime }+{\alpha }_2^{\prime }.$ we obtain that

$$\begin{array}{c}\hfill {(\Im -{\varsigma }_0)}^{{}^{\frac{p-1}{p}}}{(\varsigma -{\varsigma }_0)}^{{}^{\frac{p_{*}-1}{p_{*}}}}⩾{\left(\frac{p{p}_{*}}{p+p_{*}}\left({\displaystyle \frac{{(\Im -{\varsigma }_0)}^{p-1}}{p}}+{\displaystyle \frac{{(\varsigma -{\varsigma }_0)}^{p_{*}-1}}{p_{*}}}\right)\right)}^{\frac{p+p_{*}}{p{p}_{*}}}.\end{array}$$

(20)

Substituting (20) in (18), yields

$$\begin{array}{c}\hfill {\psi }^K(\Im ){\phi }^L\left(\varsigma \right)⩾\frac{KL}{{(p+p_{*})}^{\frac{p+p_{*}}{p{p}_{*}}}}{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}\\ \hfill \times {\left({\int }_{{\varsigma }_0}^{\Im }{\left(a\left(\eta \right){\psi }^{K-1}\left(\eta \right)\right)}^p\Delta \eta \right)}^{\frac{1}{p}}\\ \hfill \times {\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(b\left(\eta \right){\phi }^{L-1}\left(\eta \right)\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}.\end{array}$$

(21)

Dividing both side of (21) by ${\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}$, we obtain that

$$\begin{array}{c}\hfill \frac{{\psi }^K(\Im ){\phi }^L\left(\varsigma \right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}⩾\frac{KL}{{(p+p_{*})}^{\frac{p+p_{*}}{p{p}_{*}}}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left(a\left(\eta \right){\psi }^{K-1}\left(\eta \right)\right)}^p\Delta \eta \right)}^{\frac{1}{p}}\\ \hfill \times {\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(b\left(\eta \right){\phi }^{L-1}\left(\eta \right)\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}.\end{array}$$

(22)

Integrating both sides of (22) from ${\varsigma }_0$ to $\xi$ and from ${\varsigma }_0$ to $\zeta$, and applying inverse Hölder’s inequality with indices $p,$ ${\displaystyle \frac{p}{p-1}}$ and $p_{*},$${\displaystyle \frac{p_{*}}{p_{*}-1}},$ we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }\frac{{\psi }^K(\Im ){\phi }^L\left(\varsigma \right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾\frac{KL}{{(p+p_{*})}^{\frac{p+p_{*}}{p{p}_{*}}}}{(\xi -{\varsigma }_0)}^{\frac{p-1}{p}}{(\zeta -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}{\left({\int }_{{\varsigma }_0}^{\xi }\left({\int }_{{\varsigma }_0}^{\Im }{\left(a\left(\eta \right){\psi }^{K-1}\left(\eta \right)\right)}^p\Delta \eta \right)\Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\zeta }\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(b\left(\eta \right){\phi }^{L-1}\left(\eta \right)\right)}^{p_{*}}\Delta \eta \right)\Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

(23)

Applying Lemma 4 on the right hand side of (23), we have

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }\frac{{\psi }^K(\Im ){\phi }^L\left(\varsigma \right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾\frac{KL}{{(p+p_{*})}^{\frac{p+p_{*}}{p{p}_{*}}}}{(\xi -{\varsigma }_0)}^{\frac{p-1}{p}}{(\zeta -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}{\left({\int }_{{\varsigma }_0}^{\xi }(\xi -\sigma (\Im )){\left(a(\Im ){\psi }^{K-1}(\Im )\right)}^p\Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\zeta }(\zeta -\sigma \left(\varsigma \right)){\left(b\left(\varsigma \right){\phi }^{L-1}\left(\varsigma \right)\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}}\hfill \\ & & =C_1(K,L,p,p_{*}){\left({\int }_{{\varsigma }_0}^{\xi }(\xi -\sigma (\Im )){\left(a(\Im ){\psi }^{K-1}(\Im )\right)}^p\Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\zeta }(\zeta -\sigma \left(\varsigma \right)){\left(b\left(\varsigma \right){\phi }^{L-1}\left(\varsigma \right)\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

By using the facts $\xi ⩾\rho \left(\xi \right)$ and $\zeta ⩾\rho \left(\zeta \right),$ we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }\frac{{\psi }^K(\Im ){\phi }^L\left(\varsigma \right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾C_1(K,L,p,p_{*}){\left({\int }_{{\varsigma }_0}^{\xi }(\rho \left(\xi \right)-\sigma (\Im )){\left(a(\Im ){\psi }^{K-1}(\Im )\right)}^p\Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\zeta }(\rho \left(\zeta \right)-\sigma \left(\varsigma \right)){\left(b\left(\varsigma \right){\phi }^{L-1}\left(\varsigma \right)\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

This completes the proof. □

Theorem 4.

Let $a\left(\tau \right),$ $b\left(\eta \right),$ $\psi (\Im )$ and $\phi \left(\varsigma \right)$ be defined as Theorem 3. Then, we have

$$\begin{array}{ccc}& & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }\frac{\psi (\Im )\phi \left(\varsigma \right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}\Delta \Im \Delta \varsigma \hfill \\ & & ⩾{\displaystyle \frac{{(\xi -{\varsigma }_0)}^{\frac{p-1}{p}}{(\zeta -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}}{{(p+p_{*})}^{\frac{p+p_{*}}{p{p}_{*}}}}}{\left({\int }_{{\varsigma }_0}^{\xi }(\rho \left(\xi \right)-\sigma (\Im )){\left(a(\Im )\right)}^p\Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\zeta }(\rho \left(\zeta \right)-\sigma \left(\varsigma \right)){\left(b\left(\varsigma \right)\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

Proof. 

Put $K=L=1$ in (13). This completes the proof. □

As a special case of Theorem 3, when $\mathbb{T}=\mathbb{R}$, we have $\rho \left(\xi \right)=\xi ,$ $\rho \left(\zeta \right)=\zeta ,$ $\sigma (\Im )=\Im ,$ $\sigma \left(\varsigma \right)=\varsigma$, and we obtain the following result:

Corollary 1.

Assume that $a(\Im )$ and $b\left(\varsigma \right)$ are non-negative functions and define $\psi (\Im ):={\int }_0^{\Im }a\left(\eta \right)d\eta$ and $\phi \left(\varsigma \right):={\int }_0^{\varsigma }b\left(\eta \right)d\eta .$ Then,

$$\begin{array}{ccc}\hfill & & {\int }_0^{\xi }{\int }_0^{\zeta }\frac{{\psi }^K(\Im ){\phi }^L\left(\varsigma \right)}{{\left(p_{*}{\Im }^{p-1}+p{\varsigma }^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}d\Im d\varsigma \hfill \\ \hfill & & ⩾C_3(K,L,p,p_{*}){\left({\int }_0^{\xi }(\xi -\Im ){\left(a(\Im ){\psi }^{K-1}(\Im )\right)}^{p}d\Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_0^{\zeta }(\zeta -\varsigma ){\left(b\left(\varsigma \right){\phi }^{L-1}\left(\varsigma \right)\right)}^{p_{*}}d\varsigma \right)}^{\frac{1}{p_{*}}},\hfill \end{array}$$

where

$$C_3(L,K,p,p_{*})={\displaystyle \frac{KL{\xi }^{\frac{p-1}{p}}{\zeta }^{\frac{p_{*}-1}{p_{*}}}}{{(p+p_{*})}^{\frac{p+p_{*}}{p{p}_{*}}}}}.$$

As special case of Theorem 3, when $\mathbb{T}=\mathbb{Z}$, we have $\rho \left(\xi \right)=\xi -1,$ $\rho \left(\zeta \right)=\zeta -1,$ $\sigma (\Im )=\Im +1,$ $\sigma \left(\varsigma \right)=\varsigma +1$, and we obtain the following result:

Corollary 2.

Assume that $a\left(n\right)$ and $b\left(m\right)$ are non-negative sequences and define

$$\psi \left(n\right)=\sum _{\Im =0}^{n}a(\Im )\mathit{and}\phi \left(m\right)=\sum _{k=0}^{m}b\left(k\right).$$

Then,

$$\begin{array}{c}\hfill \sum _{n=1}^N\sum _{m=1}^M{\displaystyle \frac{{\psi }^L\left(n\right){\phi }^K\left(m\right)}{{\left(p_{*}{n}^{p-1}+p{m}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}⩾C_4(K,L,p,p_{*}){\left(\sum _{n=1}^N((N-1)-(n+1)){\left(a\left(n\right){\psi }^{L-1}\left(n\right)\right)}^p\right)}^{\frac{1}{p}}\\ \hfill \times {\left(\sum _{m=1}^M((M-1)-(m+1)){\left(b\left(m\right){\phi }^{L-1}\left(m\right)\right)}^{p_{*}}\right)}^{\frac{1}{p_{*}}},\end{array}$$

where

$$C_4(K,L,p)={\displaystyle \frac{KL{N}^{\frac{p-1}{p}}{M}^{\frac{p_{*}-1}{p_{*}}}}{{(p+p_{*})}^{\frac{p+p_{*}}{p{p}_{*}}}}}.$$

Theorem 5.

Let $\mathbb{T}$ be a time scale with $\Im ,$ $\varsigma ,$ ${\varsigma }_0,$ $\xi ,$ $\zeta \in \mathbb{T},$ $\psi (\Im )$ and $\phi \left(\varsigma \right)$ be as defined in Theorem 3. Let $f\left(\tau \right)$ and $g\left(\eta \right)$ be two non-negative and right-dense continuous functions on $[{\varsigma }_0,\xi ]$ and $[{\varsigma }_0,\zeta ]$, respectively. Suppose that Λ and Υ are non-negative, concave and supermultiplicative functions defined on $[0,\infty )$. Furthermore, assume that

$$\Theta (\Im ):={\int }_{{\varsigma }_0}^{\Im }f\left(\tau \right)\Delta \tau and\Xi \left(\varsigma \right):={\int }_{{\varsigma }_0}^{\varsigma }g\left(\eta \right)\Delta \eta ,$$

(24)

then, for $\Im \in [{\varsigma }_0,\xi ]$ and $\varsigma \in [{\varsigma }_0,\zeta ],$ we have that

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ & & ⩾M_1\left(p\right){\left({\int }_{{\varsigma }_0}^{\xi }(\rho \left(\xi \right)-\sigma (\Im )){\left(f(\Im )\Lambda \left[\frac{a(\Im )}{f(\Im )}\right]\right)}^p\Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\zeta }(\rho \left(\zeta \right)-\sigma \left(\varsigma \right)){\left(g\left(\varsigma \right){\rm Y}\left[\frac{b\left(\varsigma \right)}{g\left(\varsigma \right)}\right]\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}},\hfill \end{array}$$

(25)

where

$$M_1\left(p\right)={\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{\left\{{\int }_{{\varsigma }_0}^{\xi }{\left({\displaystyle \frac{\Lambda (\Theta (\Im )}{\Theta (\Im )}}\right)}^{\frac{p}{p-1}}\Delta \Im \right\}}^{\frac{p-1}{p}}{\left\{{\int }_{{\varsigma }_0}^{\zeta }{\left({\displaystyle \frac{{\rm Y}(\Xi (\varsigma \left)\right)}{\Xi \left(\varsigma \right)}}\right)}^{\frac{p_{*}}{p_{*}-1}}\Delta \varsigma \right\}}^{\frac{p_{*}-1}{p_{*}}}.$$

Proof. 

Since $\Lambda$ is a concave and supermultiplicative function, we obtain by applying inverse Jensen’s inequality that

$$\begin{array}{ccc}\hfill \Lambda \left(\psi \right(\Im \left)\right)& =& \Lambda \left({\displaystyle \frac{\Theta (\Im )){\int }_{{\varsigma }_0}^{\Im }f\left(\tau \right)\frac{a\left(\tau \right)}{f\left(\tau \right)}\Delta \tau }{{\int }_{{\varsigma }_0}^{\Im }f\left(\tau \right)\Delta \tau }}\right)\hfill \\ & ⩾& \Lambda (\Theta (\Im ))\Lambda \left({\displaystyle \frac{{\int }_{{\varsigma }_0}^{\Im }f\left(\tau \right)\frac{a\left(\tau \right)}{f\left(\tau \right)}\Delta \tau }{{\int }_{{\varsigma }_0}^{\Im }f\left(\tau \right)\Delta \tau }}\right)\hfill \\ & ⩾& {\displaystyle \frac{\Lambda (\Theta (\Im \left)\right)}{\Theta (\Im )}}{\int }_{{\varsigma }_0}^{\Im }f\left(\tau \right)\Lambda \left({\displaystyle \frac{a\left(\tau \right)}{f\left(\tau \right)}}\right)\Delta \tau .\hfill \end{array}$$

(26)

Applying inverse Hölder’s inequality with indices p and $\frac{p}{p-1}$ on the right hand side of (26), we see that

$$\Lambda \left(\psi (\Im )\right)⩾{\displaystyle \frac{\Lambda (\Theta (\Im )}{\Theta (\Im )}}{(\Im -{\varsigma }_0)}^{\frac{p-1}{p}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[\frac{a\left(\tau \right)}{f\left(\tau \right)}\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}.$$

(27)

Moreover, since ${\rm Y}$ is a concave and supermultiplicative function, we obtain by applying inverse Jensen’s inequality and inverse Hölder’s inequality with indices $p_{*}$ and $\frac{p_{*}}{p_{*}-1}$ that we have

$${\rm Y}\left(\phi \left(\varsigma \right)\right)⩾{\displaystyle \frac{{\rm Y}(\Xi (\varsigma \left)\right)}{\Xi \left(\varsigma \right)}}{(\varsigma -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}.$$

(28)

From (27) and (28), we have

$$\begin{array}{ccc}\hfill \Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)& ⩾& {(\Im -{\varsigma }_0)}^{\frac{p-1}{p}}{(\varsigma -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}\left({\displaystyle \frac{\Lambda (\Theta (\Im )}{\Theta (\Im )}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[\frac{a\left(\tau \right)}{f\left(\tau \right)}\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}\right)\hfill \\ & & \times \left({\displaystyle \frac{{\rm Y}(\Xi (\varsigma \left)\right)}{\Xi \left(\varsigma \right)}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}\right).\hfill \end{array}$$

(29)

By using inequality (19), we obtain that

$$\begin{array}{c}\hfill {(\Im -{\varsigma }_0)}^{{}^{\frac{p-1}{p}}}{(\varsigma -{\varsigma }_0)}^{{}^{\frac{p_{*}-1}{p_{*}}}}⩾{\left(\frac{p{p}_{*}}{p+p_{*}}\left({\displaystyle \frac{{(\Im -{\varsigma }_0)}^{p-1}}{p}}+{\displaystyle \frac{{(\varsigma -{\varsigma }_0)}^{p_{*}-1}}{p_{*}}}\right)\right)}^{\frac{p+p_{*}}{p{p}_{*}}}.\end{array}$$

(30)

From (29) and (30), we have that

$$\begin{array}{ccc}\hfill & & \Lambda \left(\psi (\Im )\right){\rm Y}\left(\phi \left(\varsigma \right)\right)⩾{\left(\frac{p{p}_{*}}{p+p_{*}}\left({\displaystyle \frac{{(\Im -{\varsigma }_0)}^{p-1}}{p}}+{\displaystyle \frac{{(\varsigma -{\varsigma }_0)}^{p_{*}-1}}{p_{*}}}\right)\right)}^{\frac{p+p_{*}}{p{p}_{*}}}\hfill \\ \hfill & & \times \left({\displaystyle \frac{\Lambda (\Theta (\Im )}{\Theta (\Im )}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[\frac{a\left(\tau \right)}{f\left(\tau \right)}\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}\right)\left({\displaystyle \frac{{\rm Y}(\Xi (\varsigma \left)\right)}{\Xi \left(\varsigma \right)}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}\right).\hfill \end{array}$$

(31)

Then,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}& ⩾& {\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}\left({\displaystyle \frac{\Lambda (\Theta (\Im )}{\Theta (\Im )}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[\frac{a\left(\tau \right)}{f\left(\tau \right)}\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}\right)\hfill \\ & & \times \left({\displaystyle \frac{{\rm Y}(\Xi (\varsigma \left)\right)}{\Xi \left(\varsigma \right)}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}\right).\hfill \end{array}$$

(32)

Integrating both sides of (32) from ${\varsigma }_0$ to $\xi$ and from ${\varsigma }_0$ and $\zeta$, we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ & & ⩾{\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{\int }_{{\varsigma }_0}^{\xi }\left({\displaystyle \frac{\Lambda (\Theta (\Im )}{\Theta (\Im )}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[\frac{a\left(\tau \right)}{f\left(\tau \right)}\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}\Delta \Im \right)\hfill \\ & & \times {\int }_{{\varsigma }_0}^{\zeta }\left({\displaystyle \frac{{\rm Y}(\Xi (\varsigma \left)\right)}{\Xi \left(\varsigma \right)}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}\Delta \varsigma \right).\hfill \end{array}$$

(33)

Applying inverse Hölder’s inequality with indices $p,$$\frac{p}{p-1}$ and $p_{*},$$\frac{p_{*}-1}{p_{*}}$ on the right hand of side (33), we have

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ & & ⩾{\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{\left\{{\int }_{{\varsigma }_0}^{\xi }{\left({\displaystyle \frac{\Lambda (\Theta (\Im )}{\Theta (\Im )}}\right)}^{\frac{p}{p-1}}\Delta \Im \right\}}^{\frac{p-1}{p}}{\left({\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[\frac{a\left(\tau \right)}{f\left(\tau \right)}\right]\right)}^p\Delta \tau \Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left\{{\int }_{{\varsigma }_0}^{\zeta }{\left({\displaystyle \frac{{\rm Y}(\Xi (\varsigma \left)\right)}{\Xi \left(\varsigma \right)}}\right)}^{\frac{p_{*}}{p_{*}-1}}\Delta \varsigma \right\}}^{\frac{p_{*}-1}{p_{*}}}{\left({\int }_{{\varsigma }_0}^{\zeta }{\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[\frac{b\left(\eta \right)}{g\left(\eta \right)}\right]\right)}^{p_{*}}\Delta \eta \Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

(34)

Applying Lemma 4 on the right hand side of (34), we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾M_1\left(p\right){\left({\int }_{{\varsigma }_0}^{\xi }(\xi -\sigma (\Im )){\left(f(\Im )\Lambda \left[\frac{a(\Im )}{f(\Im )}\right]\right)}^p\Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\zeta }(\zeta -\sigma \left(\varsigma \right)){\left(g\left(\varsigma \right){\rm Y}\left[\frac{b\left(\varsigma \right)}{g\left(\varsigma \right)}\right]\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

By using the facts $\xi ⩾\rho \left(\xi \right)$ and $\zeta ⩾\rho \left(\zeta \right),$ we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾M_1\left(p\right){\left({\int }_{{\varsigma }_0}^{\xi }(\rho \left(\xi \right)-\sigma (\Im )){\left(f(\Im )\Lambda \left[\frac{a(\Im )}{f(\Im )}\right]\right)}^p\Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\zeta }(\rho \left(\zeta \right)-\sigma \left(\varsigma \right)){\left(g\left(\varsigma \right){\rm Y}\left[\frac{b\left(\varsigma \right)}{g\left(\varsigma \right)}\right]\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}},\hfill \end{array}$$

where

$$M_1\left(p\right)={\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{\left\{{\int }_{{\varsigma }_0}^{\xi }{\left({\displaystyle \frac{\Lambda (\Theta (\Im )}{\Theta (\Im )}}\right)}^{\frac{p}{p-1}}\Delta \Im \right\}}^{\frac{p-1}{p}}{\left\{{\int }_{{\varsigma }_0}^{\zeta }{\left({\displaystyle \frac{{\rm Y}(\Xi (\varsigma \left)\right)}{\Xi \left(\varsigma \right)}}\right)}^{\frac{p_{*}}{p_{*}-1}}\Delta \varsigma \right\}}^{\frac{p_{*}-1}{p_{*}}}.$$

This completes the proof. □

As a special case of Theorem 5, when $\mathbb{T}=\mathbb{R}$, we have $\rho \left(\xi \right)=\xi$ $\rho \left(\zeta \right)=\zeta ,$ $\sigma (\Im )=\Im ,$ $\sigma \left(\varsigma \right)=\varsigma$, and we obtain the following result:

Corollary 3.

Assume that $a(\Im ),$ $b\left(\varsigma \right),$ $f\left(\tau \right)$ and $g\left(\eta \right)$ are non-negative functions and define

$$\psi (\Im ):={\int }_0^{\Im }a\left(\eta \right)d\eta ,\phi \left(\varsigma \right):={\int }_0^{\varsigma }b\left(\eta \right)d\eta ,\Theta (\Im ):={\int }_0^{\Im }f\left(\tau \right)d\tau ,and\Xi \left(\varsigma \right):={\int }_0^{\varsigma }g\left(\eta \right)d\eta .$$

Then,

$$\begin{array}{ccc}\hfill & & {\int }_0^{\xi }{\int }_0^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)}{{\left(p_{*}{\Im }^{p-1}+p{\varsigma }^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}d\Im d\varsigma \hfill \\ \hfill & & ⩾M_2\left(p\right){\left({\int }_0^{\xi }(\xi -\Im ){\left(f(\Im )\Lambda \left[\frac{a(\Im )}{f(\Im )}\right]\right)}^{p}d\Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_0^{\zeta }(\zeta -\varsigma ){\left(g\left(\varsigma \right){\rm Y}\left[\frac{b\left(\varsigma \right)}{g\left(\varsigma \right)}\right]\right)}^{p_{*}}d\varsigma \right)}^{\frac{1}{p_{*}}},\hfill \end{array}$$

where

$$M_2\left(p\right)={\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{\left\{{\int }_0^{\xi }{\left({\displaystyle \frac{\Lambda (\Theta (\Im )}{\Theta (\Im )}}\right)}^{\frac{p}{p-1}}d\Im \right\}}^{\frac{p-1}{p}}{\left\{{\int }_0^{\zeta }{\left({\displaystyle \frac{{\rm Y}(\Xi (\varsigma \left)\right)}{\Xi \left(\varsigma \right)}}\right)}^{\frac{p_{*}}{p_{*}-1}}d\varsigma \right\}}^{\frac{p_{*}-1}{p_{*}}}.$$

As a special case of Theorem 5, when $\mathbb{T}=\mathbb{Z}$, we have $\rho \left(\xi \right)=\xi -1,$ $\rho \left(\zeta \right)=\zeta -1,$ $\sigma (\Im )=\Im +1,$ $\sigma \left(\varsigma \right)=\varsigma +1$, and we obtain the following result.

Corollary 4.

Assume that $a\left(n\right),$ $b\left(m\right),$ $f\left(n\right)$ and $g\left(m\right)$ are non-negative sequences and define

$$\psi \left(n\right)=\sum _{\Im =0}^{n}a(\Im ),\phi \left(m\right)=\sum _{k=0}^{m}b\left(k\right),\Theta \left(n\right)=\sum _{\Im =0}^{n}f(\Im )\mathit{and}\Xi \left(m\right)=\sum _{k=0}^{m}g\left(k\right).$$

Then,

$$\begin{array}{c}\hfill \sum _{n=1}^N\sum _{m=1}^M{\displaystyle \frac{\Lambda \left(\psi \right(n\left)\right){\rm Y}\left(\phi \right(m\left)\right)}{{\left(p_{*}{n}^{p-1}+p{m}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}⩾M_3\left(p\right){\left\{\sum _{n=1}^N((N-1)-(n+1)){\left(f\left(n\right)\Lambda \left[\frac{a\left(n\right)}{f\left(n\right)}\right]\right)}^p\right\}}^{\frac{1}{p}}\\ \hfill \times {\left\{\sum _{m=1}^M((M-1)-(m+1)){\left(g\left(m\right){\rm Y}\left[\frac{b\left(m\right)}{g\left(m\right)}\right]\right)}^{p_{*}}\right\}}^{\frac{1}{p_{*}}},\end{array}$$

where

$$M_3\left(p\right)={\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{\left\{\sum _{n=1}^N{\left({\displaystyle \frac{\Lambda (\Theta (n)}{\Theta \left(n\right)}}\right)}^{\frac{p}{p-1}}\right\}}^{\frac{p-1}{p}}{\left\{\sum _{m=1}^M{\left({\displaystyle \frac{{\rm Y}(\Xi (m\left)\right)}{\Xi \left(m\right)}}\right)}^{\frac{p_{*}}{p_{*}-1}}\right\}}^{\frac{p_{*}-1}{p_{*}}}.$$

Theorem 6.

Let $\mathbb{T}$ be a time scale with $\Im ,$ $\varsigma ,$ ${\varsigma }_0,$ $\xi ,$ $\zeta \in \mathbb{T}.$ Let f and g be two non-negative and right-dense continuous functions on $[{\varsigma }_0,\xi ]$ and $[{\varsigma }_0,\zeta ]$, respectively. Suppose that Λ and Υ are non-negative, concave and supermultiplicative functions defined on $[0,\infty )$ and define

$$\Theta (\Im ):=\frac{1}{\Im -{\varsigma }_0}{\int }_{{\varsigma }_0}^{\Im }f\left(\tau \right)\Delta \tau and\Xi \left(\varsigma \right):=\frac{1}{\varsigma -{\varsigma }_0}{\int }_{{\varsigma }_0}^{\varsigma }g\left(\tau \right)\Delta \tau ,$$

(35)

then, for $\Im \in [{\varsigma }_0,\xi ]$ and $\varsigma \in [{\varsigma }_0,\zeta ],$ we have that

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda (\Theta (\Im )){\rm Y}(\Xi \left(\varsigma \right))(\Im -{\varsigma }_0)(\varsigma -{\varsigma }_0)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ & & ⩾E(p,p_{*}){\left({\int }_{{\varsigma }_0}^{\xi }(\rho \left(\xi \right)-\sigma (\Im )){\left[\Lambda \left(f(\Im )\right)\right]}^p\Delta \Im \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\zeta }(\rho \left(\zeta \right)-\sigma \left(\varsigma \right)){\left[{\rm Y}\left(g\left(\varsigma \right)\right)\right]}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}},\hfill \end{array}$$

(36)

where

$$E(p,p_{*})={\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{(\xi -{\varsigma }_0)}^{\frac{p-1}{p}}{(\zeta -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}.$$

Proof. 

By assumption and by using the inverse Jensen inequality, we see that

$$\begin{array}{ccc}\hfill \Lambda (\Theta (\Im \left)\right)& =& \Lambda \left(\frac{1}{\Im -{\varsigma }_0}{\int }_{{\varsigma }_0}^{\Im }f\left(\tau \right)\Delta \tau \right)\hfill \\ & ⩾& \frac{1}{\Im -{\varsigma }_0}{\int }_{{\varsigma }_0}^{\Im }\Lambda \left(f\left(\tau \right)\right)\Delta \tau .\hfill \end{array}$$

(37)

By applying inverse Hölder’s inequality on (37) with indices, $p,$$\frac{p}{p-1},$ we have

$$\begin{array}{c}\hfill \Lambda (\Theta (\Im ))⩾\frac{1}{\Im -{\varsigma }_0}{(\Im -{\varsigma }_0)}^{\frac{p-1}{p}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left[\Lambda \left(f\left(\tau \right)\right)\right]}^p\Delta \tau \right)}^{\frac{1}{p}}.\end{array}$$

(38)

This implies that

$$\begin{array}{c}\hfill \Lambda (\Theta (\Im ))(\Im -{\varsigma }_0)⩾{(\Im -{\varsigma }_0)}^{\frac{p-1}{p}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left[\Lambda \left(f\left(\tau \right)\right)\right]}^p\Delta \tau \right)}^{\frac{1}{p}}.\end{array}$$

(39)

Analogously,

$$\begin{array}{c}\hfill {\rm Y}(\Xi \left(\varsigma \right))(\varsigma -{\varsigma }_0)⩾{(\varsigma -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left[{\rm Y}\left(g\left(\tau \right)\right)\right]}^{p_{*}}\Delta \tau \right)}^{\frac{1}{p_{*}}}.\end{array}$$

(40)

From (39) and (40), we obtain

$$\begin{array}{ccc}\hfill \Lambda (\Theta (\Im )){\rm Y}(\Xi \left(\varsigma \right))(\Im -{\varsigma }_0)(\varsigma -{\varsigma }_0)& ⩾& {(\Im -{\varsigma }_0)}^{\frac{p-1}{p}}{(\varsigma -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left[\Lambda \left(f\left(\tau \right)\right)\right]}^p\Delta \tau \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\varsigma }{\left[{\rm Y}\left(g\left(\tau \right)\right)\right]}^{p_{*}}\Delta \tau \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

(41)

By using inequality (19), we obtain that

$$\begin{array}{c}\hfill {(\Im -{\varsigma }_0)}^{{}^{\frac{p-1}{p}}}{(\varsigma -{\varsigma }_0)}^{{}^{\frac{p_{*}-1}{p_{*}}}}⩾{\left(\frac{p{p}_{*}}{p+p_{*}}\left({\displaystyle \frac{{(\Im -{\varsigma }_0)}^{p-1}}{p}}+{\displaystyle \frac{{(\varsigma -{\varsigma }_0)}^{p_{*}-1}}{p_{*}}}\right)\right)}^{\frac{p+p_{*}}{p{p}_{*}}}.\end{array}$$

(42)

Then,

$$\begin{array}{ccc}\hfill \Lambda (\Theta (\Im )){\rm Y}(\Xi \left(\varsigma \right))(\Im -{\varsigma }_0)(\varsigma -{\varsigma }_0)& ⩾& {\left(\frac{p{p}_{*}}{p+p_{*}}\left({\displaystyle \frac{{(\Im -{\varsigma }_0)}^{p-1}}{p}}+{\displaystyle \frac{{(\varsigma -{\varsigma }_0)}^{p_{*}-1}}{p_{*}}}\right)\right)}^{\frac{p+p_{*}}{p{p}_{*}}}\hfill \\ & & {\left({\int }_{{\varsigma }_0}^{\Im }{\left[\Lambda \left(f\left(\tau \right)\right)\right]}^p\Delta \tau \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left[{\rm Y}\left(g\left(\tau \right)\right)\right]}^{p_{*}}\Delta \tau \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

(43)

From (43), we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\Lambda (\Theta (\Im )){\rm Y}(\Xi \left(\varsigma \right))(\Im -{\varsigma }_0)(\varsigma -{\varsigma }_0)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}& ⩾& {\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left[\Lambda \left(f\left(\tau \right)\right)\right]}^p\Delta \tau \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\varsigma }{\left[{\rm Y}\left(g\left(\tau \right)\right)\right]}^{p_{*}}\Delta \tau \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

(44)

Taking delta integrating on both sides of (44), first over ℑ from ${\varsigma }_0$ to $\xi$ and then over $\varsigma$ from ${\varsigma }_0$ to $\zeta ,$ we find that

$$\begin{array}{ccc}\hfill {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda (\Theta (\Im )){\rm Y}(\Xi \left(\varsigma \right))(\Im -{\varsigma }_0)(\varsigma -{\varsigma }_0)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma & ⩾& {\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}\left({\int }_{{\varsigma }_0}^{\xi }{\left({\int }_{{\varsigma }_0}^{\Im }{\left[\Lambda \left(f\left(\tau \right)\right)\right]}^p\Delta \tau \right)}^{\frac{1}{p}}\Delta \Im \right)\hfill \\ & & \times \left({\int }_{{\varsigma }_0}^{\zeta }{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left[{\rm Y}\left(g\left(\tau \right)\right)\right]}^{p_{*}}\Delta \tau \right)}^{\frac{1}{p_{*}}}\Delta \varsigma \right).\hfill \end{array}$$

(45)

By applying inverse Hölder’sinequality on (45) with indices $p,$ ${\displaystyle \frac{p}{p-1}}$ and $p_{*},$${\displaystyle \frac{p_{*}}{p_{*}-1}},$ we get

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda (\Theta (\Im )){\rm Y}(\Xi \left(\varsigma \right))(\Im -{\varsigma }_0)(\varsigma -{\varsigma }_0)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma ⩾{\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{(\xi -{\varsigma }_0)}^{\frac{p-1}{p}}{(\zeta -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\xi }\left({\int }_{{\varsigma }_0}^{\Im }{\left[\Lambda \left(f\left(\tau \right)\right)\right]}^p\Delta \tau \right)\Delta \Im \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\zeta }\left({\int }_{{\varsigma }_0}^{\varsigma }{\left[{\rm Y}\left(g\left(\tau \right)\right)\right]}^{p_{*}}\Delta \tau \right)\Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

(46)

Applying Lemma 4 on (46), we fined that

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda (\Theta (\Im )){\rm Y}(\Xi \left(\varsigma \right))(\Im -{\varsigma }_0)(\varsigma -{\varsigma }_0)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma ⩾{\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{(\xi -{\varsigma }_0)}^{\frac{p-1}{p}}{(\zeta -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}\hfill \\ \hfill & & \times {\left({\int }_{{\varsigma }_0}^{\xi }(\xi -\sigma (\Im )){\left[\Lambda \left(f(\Im )\right)\right]}^p\Delta \Im \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\zeta }(\zeta -\sigma \left(\varsigma \right)){\left[{\rm Y}\left(g\left(\varsigma \right)\right)\right]}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}}\hfill \\ & & =E(p,p_{*}){\left({\int }_{{\varsigma }_0}^{\xi }(\xi -\sigma (\Im )){\left[\varphi \left(f(\Im )\right)\right]}^p\Delta \Im \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\zeta }(\zeta -\sigma \left(\varsigma \right)){\left[{\rm Y}\left(g\left(\varsigma \right)\right)\right]}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

By using the facts $\xi ⩾\rho \left(\xi \right)$ and $\zeta ⩾\rho \left(\zeta \right),$ we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda (\Theta (\Im )){\rm Y}(\Xi \left(\varsigma \right))(\Im -{\varsigma }_0)(\varsigma -{\varsigma }_0)}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ & & ⩾E(p,p_{*}){\left({\int }_{{\varsigma }_0}^{\xi }(\rho \left(\xi \right)-\sigma (\Im )){\left[\Lambda \left(f(\Im )\right)\right]}^p\Delta \Im \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\zeta }(\rho \left(\zeta \right)-\sigma \left(\varsigma \right)){\left[{\rm Y}\left(g\left(\varsigma \right)\right)\right]}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}},\hfill \end{array}$$

where

$$E(p,p_{*})={\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{(\xi -{\varsigma }_0)}^{\frac{p-1}{p}}{(\zeta -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}.$$

This completes the proof. □

As a special case of Theorem 6, when $\mathbb{T}=\mathbb{R},$ we obtain the following conclusion.

Corollary 5.

Assume that f and g are non-negative functions and define

$$\Theta (\Im ):=\frac{1}{\Im }{\int }_0^{\Im }f\left(\tau \right)d\tau and\Xi \left(\varsigma \right):=\frac{1}{\varsigma }{\int }_0^{\varsigma }g\left(\tau \right)d\tau ,$$

then, for $\Im \in [0,\xi ]$ and $\varsigma \in [0,\zeta ],$ we have that

$$\begin{array}{ccc}\hfill {\int }_0^{\xi }{\int }_0^{\zeta }{\displaystyle \frac{\Lambda (\Theta (\Im \left)\right){\rm Y}(\Xi (\varsigma \left)\right)\Im \varsigma }{{\left(p_{*}{\Im }^{p-1}+p{\varsigma }^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}d\Im d\varsigma & ⩾& E_1(p,p_{*}){\left({\int }_0^{\xi }(\xi -\Im ){\left[\Lambda \left(f(\Im )\right)\right]}^{p}d\Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_0^{\zeta }(\zeta -\varsigma ){\left[{\rm Y}\left(g\left(\varsigma \right)\right)\right]}^{p_{*}}d\varsigma \right)}^{\frac{1}{p_{*}}},\hfill \end{array}$$

where

$$E_1(p,p_{*})={\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{\xi }^{\frac{p-1}{p}}{\zeta }^{\frac{p_{*}-1}{p_{*}}}.$$

As a special case of Theorem 6, when $\mathbb{T}=\mathbb{Z},$ we obtain the following conclusion.

Corollary 6.

Assume that $f\left(n\right)$ and $g\left(m\right)$ are two non-negative sequences of real numbers and define

$$\Theta \left(n\right):=\frac{1}{n}\sum _{\Im =1}^{n}f(\Im )and\Xi \left(m\right):=\frac{1}{m}\sum _{k=1}^{m}g\left(k\right),$$

then,

$$\begin{array}{ccc}\hfill \sum _{n=1}^N\sum _{m=1}^M{\displaystyle \frac{\Lambda (\Theta (n\left)\right){\rm Y}(\Xi (m\left)\right)nm}{{\left(p_{*}{n}^{p-1}+p{m}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}& ⩾& E_2(p,p_{*}){\left(\sum _{n=1}^N((N-1)-(n+1)){\left[\Lambda \left(f\left(n\right)\right)\right]}^p\right)}^{\frac{1}{p}}\hfill \\ & & \times {\left(\sum _{m=1}^M((M-1)-(m+1)){\left[{\rm Y}\left(g\left(m\right)\right)\right]}^{p_{*}}\right)}^{\frac{1}{p_{*}}},\hfill \end{array}$$

where

$$E_2(p,p_{*})={\left(\frac{1}{p+p_{*}}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}{N}^{\frac{p-1}{p}}{M}^{\frac{p_{*}-1}{p_{*}}}.$$

Theorem 7.

Let $\mathbb{T}$ be a time scale with $\Im ,$ $\varsigma ,$ ${\varsigma }_0,$ $\xi ,$ $\zeta \in \mathbb{T}.$ Let $a\left(\tau \right)$ and $b\left(\tau \right)$ be two non-negative and right-dense continuous functions on $[{\varsigma }_0,\xi ]$ and $[{\varsigma }_0,\zeta ]$, respectively. Let $\Theta ,$ $\Xi ,$ $f,$$g,$ Λ and Υ be as assumed in Theorem 5. Furthermore, assume that

$$\psi (\Im ):=\frac{1}{\Theta (\Im )}{\int }_{{\varsigma }_0}^{\Im }a\left(\tau \right)f\left(\tau \right)\Delta \tau and\phi \left(\varsigma \right):=\frac{1}{\Xi \left(\varsigma \right)}{\int }_{{\varsigma }_0}^{\varsigma }b\left(\eta \right)g\left(\eta \right)\Delta \eta ,$$

(47)

then, for $\Im \in [{\varsigma }_0,\xi ]$ and $\varsigma \in [{\varsigma }_0,\zeta ],$ we have that

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)\Xi \left(\varsigma \right)\Theta (\Im )}{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right){)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾D_1\left(p\right){\left({\int }_{{\varsigma }_0}^{\xi }(\xi -\sigma (\Im )){\left(f(\Im )\Lambda \left[a\right(\Im \left)\right]\right)}^p\Delta \Im \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\zeta }(\zeta -\sigma \left(\varsigma \right)){\left(g\left(\varsigma \right){\rm Y}\left[b\right(\varsigma \left)\right]\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}},\hfill \end{array}$$

where

$$D_1\left(p\right)=(\frac{1}{p+p_{*}}{)}^{\frac{p+p_{*}}{p{p}_{*}}}{(\xi -{\varsigma }_0)}^{\frac{p-1}{p}}{(\zeta -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}.$$

Proof. 

From (47), we see that

$$\Lambda \left(\psi (\Im )\right)=\Lambda \left(\frac{1}{\Theta (\Im )}{\int }_{{\varsigma }_0}^{\Im }f\left(\tau \right)a\left(\tau \right)\Delta \tau \right).$$

(48)

Applying inverse Hölder’s inequality with indices p and $\frac{p}{p-1}$ on the right hand side of (48), we obtain

$$\Lambda \left(\psi (\Im )\right)⩾\frac{{(\Im -{\varsigma }_0)}^{\frac{p-1}{p}}}{\Theta (\Im )}{\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[a\right(\tau \left)\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}.$$

(49)

From (49), we obtain that

$$\Lambda \left(\psi (\Im )\right)\Theta (\Im )⩾{(\Im -{\varsigma }_0)}^{\frac{p-1}{p}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[a\right(\tau \left)\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}.$$

(50)

Similarly, we obtain

$${\rm Y}\left(\phi \left(\varsigma \right)\right)\Xi \left(\varsigma \right)⩾{(\varsigma -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[b\right(\eta \left)\right]\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}.$$

(51)

From (50) and (51), we observe that

$$\begin{array}{ccc}\hfill & & \Lambda \left(\psi (\Im )\right){\rm Y}\left(\phi \left(\varsigma \right)\right)\Xi \left(\varsigma \right)\Theta (\Im )⩾{(\Im -{\varsigma }_0)}^{\frac{p-1}{p}}{(\varsigma -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[a\right(\tau \left)\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[b\right(\eta \left)\right]\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

(52)

Applying the inequality (19) on the term ${(\Im -{\varsigma }_0)}^{\frac{p-1}{p}}{(\varsigma -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}},$ we obtain the following inequality

$$\begin{array}{ccc}\hfill \Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)\Xi \left(\varsigma \right)\Theta (\Im )& ⩾& {\left(\frac{p{p}_{*}}{p+p_{*}}\left({\displaystyle \frac{{(\Im -{\varsigma }_0)}^{p-1}}{p}}+{\displaystyle \frac{{(\varsigma -{\varsigma }_0)}^{p_{*}-1}}{p_{*}}}\right)\right)}^{\frac{p+p_{*}}{p{p}_{*}}}\hfill \\ \hfill & & \times {\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[a\right(\tau \left)\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[b\right(\eta \left)\right]\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

(53)

Dividing both sides of (53) by ${\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}},$ we obtain that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)\Xi \left(\varsigma \right)\Theta (\Im )}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}& & ⩾(\frac{1}{p+p_{*}}{)}^{\frac{p+p_{*}}{p{p}_{*}}}{\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[a\right(\tau \left)\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[b\right(\eta \left)\right]\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

(54)

Integrating both sides of (54) from ${\varsigma }_0$ to $\xi$ and ${\varsigma }_0$ to $\zeta ,$ we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)\Xi \left(\varsigma \right)\Theta (\Im )}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾(\frac{1}{p+p_{*}}{)}^{\frac{p+p_{*}}{p{p}_{*}}}\left({\int }_{{\varsigma }_0}^{\xi }{\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[a\right(\tau \left)\right]\right)}^p\Delta \tau \right)}^{\frac{1}{p}}\Delta \Im \right)\left({\int }_{{\varsigma }_0}^{\zeta }{\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[b\right(\eta \left)\right]\right)}^{p_{*}}\Delta \eta \right)}^{\frac{1}{p_{*}}}\Delta \varsigma \right).\hfill \end{array}$$

(55)

Applying inverse Hölder’s inequality again with indices $p,$${\displaystyle \frac{p}{p-1}}$ and $p_{*},$${\displaystyle \frac{p_{*}}{p_{*}-1}}$ on the right hand side of (55), we have

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)\Xi \left(\varsigma \right)\Theta (\Im )}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾(\frac{1}{p+p_{*}}{)}^{\frac{p+p_{*}}{p{p}_{*}}}{(\xi -{\varsigma }_0)}^{\frac{p-1}{p}}{(\zeta -{\varsigma }_0)}^{\frac{p_{*}-1}{p_{*}}}{\left({\int }_{{\varsigma }_0}^{\xi }\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[a\right(\tau \left)\right]\right)}^p\Delta \tau \right)\Delta \Im \right)}^{\frac{1}{p}}\hfill \\ & & \times {\left({\int }_{{\varsigma }_0}^{\zeta }\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[b\right(\eta \left)\right]\right)}^{p_{*}}\Delta \eta \right)\Delta \varsigma \right)}^{\frac{1}{p_{*}}}\hfill \\ & & =D_1\left(p\right){\left({\int }_{{\varsigma }_0}^{\xi }\left({\int }_{{\varsigma }_0}^{\Im }{\left(f\left(\tau \right)\Lambda \left[a\right(\tau \left)\right]\right)}^p\Delta \tau \right)\Delta \Im \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\zeta }\left({\int }_{{\varsigma }_0}^{\varsigma }{\left(g\left(\eta \right){\rm Y}\left[b\right(\eta \left)\right]\right)}^{p_{*}}\Delta \eta \right)\Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

(56)

Applying Lemma 4 on the right hand side of (56), we obtain that

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)\Xi \left(\varsigma \right)\Theta (\Im )}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾D_1\left(p\right){\left({\int }_{{\varsigma }_0}^{\xi }(\xi -\sigma (\Im )){\left(f(\Im )\Lambda \left[a\right(\Im \left)\right]\right)}^p\Delta \Im \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\zeta }(\zeta -\sigma \left(\varsigma \right)){\left(g\left(\varsigma \right){\rm Y}\left[b\right(\varsigma \left)\right]\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

By using the facts $\xi ⩾\rho \left(\xi \right)$ and $\zeta ⩾\rho \left(\zeta \right),$ we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{{\varsigma }_0}^{\xi }{\int }_{{\varsigma }_0}^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)\Xi \left(\varsigma \right)\Theta (\Im )}{{\left(p_{*}{(\Im -{\varsigma }_0)}^{p-1}+p{(\varsigma -{\varsigma }_0)}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}\Delta \Im \Delta \varsigma \hfill \\ \hfill & & ⩾D_1\left(p\right){\left({\int }_{{\varsigma }_0}^{\xi }(\rho \left(\xi \right)-\sigma (\Im )){\left(f(\Im )\Lambda \left[a\right(\Im \left)\right]\right)}^p\Delta \Im \right)}^{\frac{1}{p}}{\left({\int }_{{\varsigma }_0}^{\zeta }(\rho \left(\zeta \right)-\sigma \left(\varsigma \right)){\left(g\left(\varsigma \right){\rm Y}\left[b\right(\varsigma \left)\right]\right)}^{p_{*}}\Delta \varsigma \right)}^{\frac{1}{p_{*}}}.\hfill \end{array}$$

This completes the proof. □

As a special case of Theorem 7, when $\mathbb{T}=\mathbb{R}$, we have $\rho \left(\xi \right)=\xi ,$ $\rho \left(\zeta \right)=\zeta ,$ $\sigma (\Im )=\Im ,$ $\sigma \left(\varsigma \right)=\varsigma$, and we obtain the following result:

Corollary 7.

Assume that $a(\Im ),$ $b\left(\varsigma \right),$ $f(\Im )$ and $g\left(\varsigma \right)$ are non-negative functions and define

$$\psi (\Im ):=\frac{1}{\Theta (\Im )}{\int }_0^{\Im }f\left(\tau \right)a\left(\tau \right)d\tau \mathit{and}\phi \left(\varsigma \right):=\frac{1}{\Xi \left(\varsigma \right)}{\int }_0^{\varsigma }g\left(\tau \right)b\left(\tau \right)d\tau ,$$

$$\Theta (\Im ):={\int }_0^{\Im }f\left(\tau \right)d\tau \mathit{and}\Xi \left(\varsigma \right):={\int }_0^{\varsigma }g\left(\tau \right)d\tau .$$

Then,

$$\begin{array}{ccc}\hfill {\int }_0^{\xi }{\int }_0^{\zeta }{\displaystyle \frac{\Lambda \left(\psi \right(\Im \left)\right){\rm Y}\left(\phi \right(\varsigma \left)\right)\Theta (\Im )\Xi \left(\varsigma \right)}{{\left(p_{*}{\Im }^{p-1}+p{\varsigma }^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}d\Im d\varsigma & & ⩾D_2\left(p\right){\left({\int }_0^{\xi }(\xi -\Im ){\left(f(\Im )\Lambda \left(a(\Im )\right)\right)}^{p}d\Im \right)}^{\frac{1}{p}}\hfill \\ & & \times ({\int }_0^{\zeta }(\zeta -\varsigma ){{\left(g\left(\varsigma \right){\rm Y}\left(b\left(\varsigma \right)\right)\right)}^{p_{*}}d\varsigma )}^{\frac{1}{p_{*}}},\hfill \end{array}$$

where

$$D_2\left(p\right)=(\frac{1}{p+p_{*}}{)}^{\frac{p+p_{*}}{p{p}_{*}}}{\left(\xi \right)}^{\frac{p-1}{p}}{\left(\zeta \right)}^{\frac{p_{*}-1}{p_{*}}}.$$

As a special case of Theorem 7, when $\mathbb{T}=\mathbb{Z}$, we have $\rho \left(\xi \right)=\xi -1,$ $\rho \left(\zeta \right)=\zeta -1,$ $\sigma (\Im )=\Im +1,$ $\sigma \left(\varsigma \right)=\varsigma +1$, and we obtain the following result:

Corollary 8.

Assume that $a\left(n\right),$ $b\left(m\right),$ $f\left(n\right)$ and $g\left(m\right)$ are non-negative sequences and define

$$\psi \left(n\right):=\frac{1}{\Theta \left(n\right)}\sum _{\Im =0}^{n}f(\Im )a(\Im )\mathit{and}\phi \left(m\right):=\frac{1}{\Xi \left(m\right)}\sum _{k=0}^{m}g\left(k\right)b\left(k\right),$$

$$\Theta \left(n\right):=\sum _{\Im =0}^{n}f(\Im )\mathit{and}\Xi \left(m\right):=\sum _{k=0}^{m}g\left(k\right).$$

Then

$$\begin{array}{ccc}\hfill \sum _{n=1}^N\sum _{m=1}^M{\displaystyle \frac{\Lambda \left(\psi \right(n\left)\right){\rm Y}\left(\phi \right(m\left)\right)\Theta \left(n\right)\Xi \left(m\right)}{{\left(p_{*}{n}^{p-1}+p{m}^{p_{*}-1}\right)}^{\frac{p+p_{*}}{p{p}_{*}}}}}& & ⩾D_3\left(p\right){\left(\sum _{n=1}^N((N-1)-(n+1)){\left(f\left(n\right)\Lambda \left(a\left(n\right)\right)\right)}^p\right)}^{\frac{1}{p}}.\hfill \\ & & \times (\sum _{m=1}^M((M-1)-(m+1)){{\left(g\left(m\right){\rm Y}\left(b\left(m\right)\right)\right)}^{p_{*}})}^{\frac{1}{p_{*}}},\hfill \end{array}$$

where

$$D_3\left(p\right)=(\frac{1}{p+p_{*}}{)}^{\frac{p+p_{*}}{p{p}_{*}}}{\left(N\right)}^{\frac{p-1}{p}}{\left(M\right)}^{\frac{p_{*}-1}{p_{*}}}.$$

3. Conclusions and Discussion

In this article, with the help of the inverse Hölder’s inequality and inverse Jensen’s inequality on time scales, we discussed and proved several new generalizations of the integral retarded inequalities given in [3]. Moreover, we generalized a number of other inequalities to a general time scale. Finally, as a special case, we studied the discrete and continuous inequalities. As a future work, we intend to generalize these inequalities by using alpha-conformable fractional derivatives on time scales. Furthermore, we will extend these results to diamond alpha calculus.