New Weighted Opial-Type Inequalities on Time Scales for Convex Functions

Abstract

Our work is based on the multiple inequalities illustrated in 1967 by E. K. Godunova and V. I. Levin, in 1990 by Hwang and Yang and in 1993 by B. G. Pachpatte. With the help of the dynamic Jensen and Hölder inequality, we generalize a number of those inequalities to a general time scale. In addition to these generalizations, some integral and discrete inequalities will be obtained as special cases of our results.

1. Introduction

The following inequality [1] is well-known in the literature as Opial’s inequality.

Theorem 1.

If δ is an absolutely continuous function on ${[0,h]}_{\mathbb{R}}$ with $\delta \left(0\right)=\delta \left(h\right)=0$, then

$${\int }_0^h|\delta \left(t\right){\delta }^{\prime }\left(t\right)|dt\le \frac{h}{4}{\int }_0^h|{\delta }^{\prime }\left(t\right){|}^{2}dt.$$

(1)

In 1967 E. K. Godunova and V. I. Levin [2] proved the following two theorems which are a generalization of Opial’s inequality (1).

Theorem 2.

Let δ be real-valued absolutely continuous function on ${[a,b]}_{\mathbb{R}}$ with $\delta \left(a\right)=0.$ Let f be real-valued convex and increasing function on ${[0,\infty )}_{\mathbb{R}}$ with $f\left(0\right)=0.$ Then, the following inequality holds

$$\begin{array}{c}\hfill {\displaystyle {\int }_a^b{f}^{\prime }(|\delta \left(t\right)|)|{\delta }^{\prime }\left(t\right)|\mathrm{d}t⩽f\left({\int }_a^b|{\delta }^{\prime }\left(t\right)|\mathrm{d}t\right).}\end{array}$$

Theorem 3.

Let δ be real-valued absolutely continuous function on ${[a,b]}_{\mathbb{R}}$ with $\delta \left(a\right)=\delta \left(b\right)=0.$ Assume f and g are real-valued convex and increasing functions on ${[0,\infty )}_{\mathbb{R}}$ with $f\left(0\right)=0.$ Further let $p\ge 0$ on ${[a,b]}_{\mathbb{R}}$ and ${\int }_a^{b}p\left(t\right)\mathrm{d}t=1$. Then, the following inequality holds

$${\int }_a^b{f}^{\prime }(|\delta \left(t\right)|)|{\delta }^{\prime }\left(t\right)|\mathrm{d}t\le 2f\left(g^{-1}\left({\int }_a^{b}p\left(t\right)g\left(\frac{|{\delta }^{\prime }\left(t\right)|}{2p\left(t\right)}\right)\mathrm{d}t\right)\right).$$

(2)

In 1990, Hwang and Yang [3] established the following result:

Theorem 4.

Assuming $f\ge 0$ and $g\ge 0$ are continuous functions on ${[0,\infty )}_{\mathbb{R}}$ with $f\left(0\right)=0$ such that $f^{\prime }\ge 0$ and $g^{\prime }\ge 0$ exist and non-decreasing continuous functions on ${[0,\infty )}_{\mathbb{R}}$. Suppose x and y are absolutely continuous functions on ${[a,\tau ]}_{\mathbb{R}}$, and $x\left(a\right)=y\left(a\right)=0$. Then, for all $m\ge 1$, we get

$$\begin{array}{cc}\hfill {\int }_a^{\tau }& \left[{f(|\delta \left(t\right)|}^m)g^{\prime }{(|\gamma \left(t\right)|}^m)|{\gamma }^{\prime }{\left(t\right)|}^m+{g(|\gamma \left(t\right)|}^m)f^{\prime }{(|\delta \left(t\right)|}^m)|{\delta }^{\prime }{\left(t\right)|}^m\right]\mathrm{d}t\hfill \\ \hfill \le & \frac{1}{\lambda }f\left(\lambda {\int }_a^{\tau }{|{\delta }^{\prime }\left(t\right)|}^m\mathrm{d}t\right)g\left(\lambda {\int }_a^{\tau }{|{\gamma }^{\prime }\left(t\right)|}^m\mathrm{d}t\right),\hfill \end{array}$$

(3)

where $\lambda ={(\tau -a)}^{m-1}$.

S. Hilger [4], suggested time scales theory to unify discrete and continuous analysis. Continuous calculus, discrete calculus, and quantum calculus can be said as the three most common examples of calculus on time scales i.e., for continuous calculus $\mathbb{T}=\mathbb{R},$ for discrete calculus $\mathbb{T}=h\mathbb{Z}$ and for quantum calculus $\mathbb{T}=\overline{q^{\mathbb{Z}}}=\{q^z:z\in \mathbb{Z}\}\cup \left\{0\right\}$ where $q>1$. The book due to Bohner and Peterson [5] on the subject of time scales briefs and organizes much of time scales calculus. For some Opial-type integral, dynamic inequalities and other types of inequalities on time scales, see the papers [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. More results on inequalities see, [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53].

The following essential relations on some time scales such as $\mathbb{R}$, $\mathbb{Z}$, $h\mathbb{Z}$ and $\overline{q^{\mathbb{Z}}}$ will be used in the following section. Note that:

(i)

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

$$\sigma \left(t\right)=t, \mu \left(t\right)=0, {\psi }^{\Delta }\left(t\right)={\psi }^{\prime }\left(t\right), {\int }_{\alpha }^{\beta }\psi \left(t\right)\Delta t={\int }_{\alpha }^{\beta }\psi \left(t\right)dt.$$

(4)

(ii)

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

$$\sigma \left(t\right)=t+1, \mu \left(t\right)=1, {\psi }^{\Delta }\left(t\right)=\psi (t+1)-\psi \left(t\right), {\int }_{\alpha }^{\beta }\psi \left(t\right)\Delta t=\sum _{t=\alpha }^{\beta -1}\psi \left(t\right).$$

(5)

(iii)

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

$$\sigma \left(t\right)=t+h, \mu \left(t\right)=h, {\psi }^{\Delta }\left(t\right)=\frac{\psi (t+h)-\psi (t}{h}), {\int }_{\alpha }^{\beta }\psi \left(t\right)\Delta t=\sum _{t=\frac{\alpha }{h}}^{\frac{\beta }{h}-1}\psi \left(th\right)h.$$

(6)

(iv)

If $\mathbb{T}=\overline{q^{\mathbb{Z}}}$, then

$$\begin{array}{c}\sigma \left(t\right)=qt, \mu \left(t\right)=(q-1)t, {\psi }^{\Delta }\left(t\right)=\frac{\psi \left(qt\right)-\psi \left(t\right)}{(q-1)t}, {\int }_{\alpha }^{\beta }\psi \left(t\right)\Delta t=(q-1){\sum }_{t=\left({\log }_q\alpha \right)}^{\left({\log }_q\beta \right)-1}\psi \left(q^t\right)q^t.\hfill \end{array}$$

(7)

Next is Hölder’s and Jensen’s inequality:

Lemma 1

(Hölder’s inequality [5]). Let $a,b\in \mathbb{T}$. For f, $g\in C_{rd}(\mathbb{T},\mathbb{R})$, we have

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

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

Lemma 2

(Jensen’s inequality [16]). 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 nonnegative with ${\int }_a^{b}r\left(t\right)\Delta t>0$. If $\breve{\Phi }\in C_{rd}\left((c,d),\mathbb{R}\right)$ is a convex function, then

$$\breve{\Phi }\left({\displaystyle \frac{{\int }_a^{b}r\left(t\right)g\left(t\right)\Delta t}{{\int }_a^{b}r\left(t\right)\Delta t}}\right)⩽{\displaystyle \frac{{\int }_a^{b}r\left(t\right)\breve{\Phi }\left(g\left(t\right)\right)\Delta t}{{\int }_a^{b}r\left(t\right)\Delta t}}.$$

Lemma 3

(Chain rule [5]). Assume $g:\mathbb{R}\to \mathbb{R}$, is continuous, $g:\mathbb{T}\to \mathbb{R}$ is delta-differentiable on ${\mathbb{T}}^{\kappa }$, and $f:\mathbb{R}\to \mathbb{R}$ is continuously differentiable. Then there exists c in the real interval $[t,\sigma (t\left)\right]$ with

$${(f\circ g)}^{\Delta }\left(t\right)=f^{\prime }\left(g\left(c\right)\right)g^{\Delta }\left(t\right).$$

(8)

In the proofs of our results, the following inequality will be used:

$$a^k+b^k\le {(a+b)}^k\le 2^{k-1}\left(a^k+b^k\right), \mathrm{if}a\ge 0,b\ge 0\mathrm{and}k\ge 1.$$

(9)

In this article, we prove some dynamic Opial-type inequalities involving convex functions on time scales. Our results generalize some of the mentioned results of Pachpatte [54,55], E. K. Godunova and V. I. Levin [2] and Hwang and Yang [3], in time scales. Furthermore, our results extend some existing dynamic Opial-type inequalities in the literature, and give some integral and discrete inequalities as special cases.

2. Main Results

Theorem 5.

Assuming $\mathbb{T}$ is a time scale with Γ, $\alpha \in \mathbb{T}$, $g\ge 0$ and $f\ge 0$ are continuous functions on ${[0,\infty )}_{\mathbb{R}}$ with $f\left(0\right)=0$ such that $g^{\prime }\ge 0$ and $f^{\prime }\ge 0$ exist and non-decreasing continuous functions on ${[0,\infty )}_{\mathbb{R}}$. Suppose x and y are rd-continuous functions on ${[\alpha ,\Gamma ]}_{\mathbb{T}}$, and $x\left(\alpha \right)=y\left(\alpha \right)=0$. Then, for all $m\ge 1$, we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& \left[f\left({|x\left(\theta \right)|}^m\right)g^{\prime }\left({|y\left(\theta \right)|}^m\right){\left|y^{\Delta }\left(\theta \right)\right|}^m+g\left({|y\left(\theta \right)|}^m\right)f^{\prime }\left({|x\left(\theta \right)|}^m\right){\left|x^{\Delta }\left(\theta \right)\right|}^m\right]\Delta \theta \hfill \\ \hfill \le & \frac{1}{{\lambda }_1}f\left({\lambda }_1{\int }_{\alpha }^{\Gamma }{\left|x^{\Delta }\left(\theta \right)\right|}^m\Delta \theta \right)g\left({\lambda }_1{\int }_{\alpha }^{\Gamma }{\left|y^{\Delta }\left(\theta \right)\right|}^m\Delta \theta \right),\hfill \end{array}$$

(10)

where ${\lambda }_1={(\Gamma -\alpha )}^{m-1}$.

Proof. 

For $t\in {[\alpha ,\Gamma ]}_{\mathbb{T}}$. Define $\phi \left(t\right):={\int }_{\alpha }^t{\left|x^{\Delta }\left(\theta \right)\right|}^m\Delta \theta$ and $\psi \left(t\right):={\int }_{\alpha }^t{\left|y^{\Delta }\left(\theta \right)\right|}^m\Delta \theta$ so that ${\phi }^{\Delta }\left(t\right)={\left|x^{\Delta }\left(t\right)\right|}^m$ and ${\psi }^{\Delta }\left(t\right)={\left|y^{\Delta }\left(t\right)\right|}^m$. Thus,

$$\begin{array}{cc}\hfill {|x\left(t\right)|}^m=& {\left|{\int }_{\alpha }^t{x}^{\Delta }\left(\theta \right)\Delta \theta \right|}^m\le {\left({\int }_{\alpha }^t\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta \right)}^m,\hfill \\ \hfill {|y\left(t\right)|}^m=& {\left|{\int }_{\alpha }^t{y}^{\Delta }\left(\theta \right)\Delta \theta \right|}^m\le {\left({\int }_{\alpha }^t\left|y^{\Delta }\left(\theta \right)\right|\Delta \theta \right)}^m.\hfill \end{array}$$

(11)

Next, applying the dynamic Hölder’s inequality (1) on (11) with indices m and $\frac{m}{m-1}$, we get

$$\begin{array}{cc}\hfill {|x\left(t\right)|}^m\le & {(t-\alpha )}^{m-1}\left({\int }_{\alpha }^t{\left|x^{\Delta }\left(\theta \right)\right|}^m\Delta \theta \right)\hfill \\ \hfill \le & {(t-\alpha )}^{m-1}\left({\int }_{\alpha }^t{\left|x^{\Delta }\left(\theta \right)\right|}^m\Delta \theta \right)\hfill \\ \hfill \le & {\lambda }_1\phi \left(t\right).\hfill \end{array}$$

(12)

and similarly

$${|y\left(t\right)|}^m\le {\lambda }_1\psi \left(t\right).$$

(13)

Since $f\ge 0$, $f^{\prime }\ge 0$, $g\ge 0$, and $g^{\prime }\ge 0$ are non-decreasing continuous on ${[0,\infty )}_{\mathbb{R}}$. Since $\sigma \left(t\right)\ge t$, we get $g\left({\lambda }_1\psi \right)\le g^{\sigma }\left({\lambda }_1\psi \right)$, and we find that

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& \left[f\left({|x\left(\theta \right)|}^m\right)g^{\prime }\left({|y\left(\theta \right)|}^m\right){\left|y^{\Delta }\left(\theta \right)\right|}^m+g\left({|y\left(\theta \right)|}^m\right)f^{\prime }\left({|x\left(\theta \right)|}^m\right){\left|x^{\Delta }\left(\theta \right)\right|}^m\right]\Delta \theta \hfill \\ \hfill \le & {\int }_{\alpha }^{\Gamma }\left[f\left({\lambda }_1\phi \left(\theta \right)\right)g^{\prime }\left({\lambda }_1\psi \left(\theta \right)\right){\psi }^{\Delta }\left(\theta \right)+g\left({\lambda }_1\psi \left(\theta \right)\right)f^{\prime }\left({\lambda }_1\phi \left(\theta \right)\right){\phi }^{\Delta }\left(\theta \right)\right]\Delta \theta \hfill \\ \hfill \le & {\int }_{\alpha }^{\Gamma }\left[f\left({\lambda }_1\phi \left(\theta \right)\right)g^{\prime }\left({\lambda }_1\psi \left(\theta \right)\right){\psi }^{\Delta }\left(\theta \right)+g^{\sigma }\left({\lambda }_1\psi \left(\theta \right)\right)f^{\prime }\left({\lambda }_1\phi \left(\theta \right)\right){\phi }^{\Delta }\left(\theta \right)\right]\Delta \theta .\hfill \end{array}$$

(14)

From Lemma 3 for $c\in {[t,\sigma \left(t\right)]}_{\mathbb{R}}$, we get

$$\frac{1}{{\lambda }_1}{f}^{\Delta }\left({\lambda }_1\phi \right)\left(t\right)=f^{\prime }\left({\lambda }_1\phi \left(c\right)\right){\phi }^{\Delta }\left(t\right)\ge f^{\prime }\left({\lambda }_1\phi \left(t\right)\right){\phi }^{\Delta }\left(t\right),$$

(15)

and similarly

$$\frac{1}{{\lambda }_1}{g}^{\Delta }\left({\lambda }_1\psi \left(t\right)\right)\ge g^{\prime }\left({\lambda }_1\psi \left(t\right)\right){\psi }^{\Delta }\left(t\right).$$

(16)

Then, by substituting (15) and (16) into (14), we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& \left[f\left({|x\left(\theta \right)|}^m\right)g^{\prime }\left({|y\left(\theta \right)|}^m\right){\left|y^{\Delta }\left(\theta \right)\right|}^m+g\left({|y\left(\theta \right)|}^m\right)f^{\prime }\left({|x\left(\theta \right)|}^m\right){\left|x^{\Delta }\left(\theta \right)\right|}^m\right]\Delta \theta \hfill \\ \hfill \le & {\int }_{\alpha }^{\Gamma }\left[f\left({\lambda }_1\phi \left(\theta \right)\right)g^{\prime }\left({\lambda }_1\psi \left(\theta \right)\right){\psi }^{\Delta }\left(\theta \right)+g^{\sigma }\left({\lambda }_1\psi \left(\theta \right)\right)f^{\prime }\left({\lambda }_1\phi \left(\theta \right)\right){\phi }^{\Delta }\left(\theta \right)\right]\Delta \theta \hfill \\ \hfill =& \frac{1}{{\lambda }_1}{\int }_{\alpha }^{\Gamma }{\left[f\left({\lambda }_1\phi \right)g\left({\lambda }_1\psi \right)\right]}^{\Delta }\left(\theta \right)\Delta \theta \hfill \\ \hfill =& \frac{1}{{\lambda }_1}f\left({\lambda }_1{\int }_{\alpha }^{\Gamma }{\left|x^{\Delta }\left(\theta \right)\right|}^m\Delta \theta \right)g\left({\lambda }_1{\int }_{\alpha }^{\Gamma }{\left|y^{\Delta }\left(\theta \right)\right|}^m\Delta \theta \right),\hfill \end{array}$$

which is the desired inequality (10). This proves our claim. □

Remark 1.

When $\mathbb{T}=\mathbb{R}$ in Theorem 5, then, by the relation (4), we obtain Hwang and Yang inequality (3).

Corollary 1.

If we take $\mathbb{T}=h\mathbb{Z}$ in Theorem 5, then, by the relation (6), inequality (10) becomes

$$\begin{array}{cc}\hfill h\sum _{n=\frac{\alpha }{h}}^{\frac{\Gamma }{h}-1}& \left[f\left({|x\left(nh\right)|}^m\right)g^{\prime }\left({|y\left(nh\right)|}^m\right){\left|\Delta y\left(nh\right)\right|}^m+g\left({|y\left(nh\right)|}^m\right)f^{\prime }\left({|x\left(nh\right)|}^m\right){\left|\Delta x\left(nh\right)\right|}^m\right]\hfill \\ \hfill \le & \frac{1}{{\lambda }_1}f\left(h{\lambda }_1\sum _{n=\frac{\alpha }{h}}^{\frac{\Gamma }{h}-1}{\left|\Delta x\left(nh\right)\right|}^m\right)g\left(h{\lambda }_1\sum _{n=\frac{\alpha }{h}}^{\frac{\Gamma }{h}-1}{\left|\Delta y\left(nh\right)\right|}^m\right).\hfill \end{array}$$

Remark 2.

In Corollary 2, if we take $h=1$, then inequality (10) becomes

$$\begin{array}{cc}\hfill \sum _{n=\alpha }^{\Gamma -1}& \left[f\left({|x\left(n\right)|}^m\right)g^{\prime }\left({|y\left(n\right)|}^m\right){\left|\Delta y\left(n\right)\right|}^m+g\left({|y\left(n\right)|}^m\right)f^{\prime }\left({|x\left(n\right)|}^m\right){\left|\Delta x\left(n\right)\right|}^m\right]\hfill \\ \hfill \le & \frac{1}{{\lambda }_1}f\left({\lambda }_1\sum _{n=\alpha }^{\Gamma -1}{\left|\Delta x\left(n\right)\right|}^m\right)g\left({\lambda }_1\sum _{n=\alpha }^{\Gamma -1}{\left|\Delta y\left(n\right)\right|}^m\right).\hfill \end{array}$$

Corollary 2.

If we take $\mathbb{T}=\overline{q^{\mathbb{Z}}}$ in Theorem 5, then, by the relation (7), inequality (10) becomes

$$\begin{array}{cc}\hfill (q-1)\sum _{n=\left({\log }_q\alpha \right)}^{({\log }_q\Gamma )-1}& \left[f\left(|x\left(q^n\right){|}^m\right)g^{\prime }\left(|y\left(q^n\right){|}^m\right){\left|\Delta y\left(q^n\right)\right|}^m+g\left(|y\left(q^n\right){|}^m\right)f^{\prime }\left(|x\left(q^n\right){|}^m\right){\left|\Delta x\left(q^n\right)\right|}^m\right]q^n\hfill \\ \hfill \le & \frac{1}{{\lambda }_1}f\left((q-1){\lambda }_1\sum _{n=\left({\log }_q\alpha \right)}^{({\log }_q\Gamma )-1}{\left|\Delta x\left(q^n\right)\right|}^m{q}^n\right)g\left((q-1){\lambda }_1\sum _{n=\left({\log }_q\alpha \right)}^{({\log }_q\Gamma )-1}{\left|\Delta y\left(q^n\right)\right|}^m{q}^n\right).\hfill \end{array}$$

Remark 3.

If $g=m=1$, and $x=y$ in inequality (10), then, we obtain

$${\int }_{\alpha }^{\Gamma }{f}^{\prime }\left(|x(\theta )|\right)\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta \le f\left({\int }_{\alpha }^{\Gamma }\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta \right)$$

(17)

with weak conditions for f.

Remark 4.

If $\mathbb{T}=\mathbb{R}$, then inequality (17) gives E. K. Godunova and V. I. Levin’s inequality in [2].

Corollary 3.

Assuming $f\left(t\right)=t^{\frac{\ell +m}{m}}$, $g=1$, $x=y$ and $\ell \ge 0$ in Theorem 5. Then, from (10), and Lemma 1 with indices $\frac{\ell +m}{\ell }$ and $\frac{\ell +m}{m}$, we obtain

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }{|x\left(\theta \right)|}^{\ell }{\left|x^{\Delta }\left(\theta \right)\right|}^m\Delta \theta \le & \frac{m}{\ell +m}\frac{1}{{\lambda }_1}{\left({\lambda }_1{\int }_{\alpha }^{\Gamma }{\left|x^{\Delta }\left(\theta \right)\right|}^m\Delta \theta \right)}^{\frac{\ell +m}{m}}\hfill \\ \hfill \le & \frac{m}{\ell +m}{\lambda }_1^{\frac{\ell }{m}}{(\Gamma -\alpha )}^{\frac{\ell }{m}}{\int }_{\alpha }^{\Gamma }{\left|x^{\Delta }\left(\theta \right)\right|}^{\ell +m}\Delta \theta \hfill \\ \hfill =& \frac{m}{\ell +m}{(\Gamma -\alpha )}^{\ell }{\int }_{\alpha }^{\Gamma }{\left|x^{\Delta }\left(\theta \right)\right|}^{\ell +m}\Delta \theta .\hfill \end{array}$$

(18)

Corollary 4.

When $m=1$ and $\ell =1$ in (18), we obtain the following inequality

$${\int }_{\alpha }^{\Gamma }|x\left(\theta \right)|\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta \le \frac{(\Gamma -\alpha )}{2}{\int }_{\alpha }^{\Gamma }{\left|x^{\Delta }\left(\theta \right)\right|}^2\Delta \theta .$$

(19)

Corollary 5.

When $\mathbb{T}=\mathbb{R}$, $m=1$ and $\alpha =0$ in (18), we obtain the inequality of Hua [56]

$${\int }_0^{\Gamma }{|x\left(\theta \right)|}^{\ell }\left|x^{\prime }\left(\theta \right)\right|\mathrm{d}\theta \le \frac{{\Gamma }^{\ell }}{\ell +1}{\int }_0^{\Gamma }{\left|x^{\prime }\left(\theta \right)\right|}^{\ell +1}\mathrm{d}\theta .$$

(20)

Moreover, in (20), equality holds if and only if $x\left(t\right)=ct$.

Corollary 6.

If $\mathbb{T}=\mathbb{R}$, then (18), gives the inequality of Yang [57].

$${\int }_{\alpha }^{\Gamma }{|x\left(\theta \right)|}^{\ell }{\left|x^{\prime }\left(\theta \right)\right|}^m\mathrm{d}\theta \le \frac{m}{\ell +m}{(\Gamma -\alpha )}^{\ell }{\int }_{\alpha }^{\Gamma }{\left|x^{\prime }\left(\theta \right)\right|}^{\ell +m}\mathrm{d}\theta .$$

(21)

Corollary 7.

Assuming $p\ge 0$ is rd-continuous on ${[\alpha ,\Gamma ]}_{\mathbb{T}}$ with ${\int }_{\alpha }^{\Gamma }\frac{\Delta \theta }{p\left(\theta \right)}<\infty$, and let $q>0$ is non-increasing bounded on ${[\alpha ,\Gamma ]}_{\mathbb{T}}$. By using $f\left(t\right)=t^2$, $g\left(t\right)=1$, $m=1$ and $x\left(t\right)=y\left(t\right)={\int }_{\alpha }^t\sqrt{q\left(\theta \right)}\left|z^{\Delta }\left(\theta \right)\right|\Delta \theta$ and $x^{\Delta }\left(t\right)=\sqrt{q\left(t\right)}\left|z^{\Delta }\left(t\right)\right|$. Then, we get from Theorem 5, the following inequality

$$2{\int }_{\alpha }^{\Gamma }\left({\int }_{\alpha }^{\theta }\sqrt{q\left(\xi \right)}\left|x^{\Delta }\left(\xi \right)\right|\Delta \xi \right)\sqrt{q\left(\theta \right)}\left|z^{\Delta }\left(\theta \right)\right|\Delta \theta \le {\left({\int }_{\alpha }^{\Gamma }\sqrt{q\left(\theta \right)}\left|z^{\Delta }\left(\theta \right)\right|\Delta \theta \right)}^2.$$

However, since

$${\int }_{\alpha }^t\sqrt{q\left(\theta \right)}\left|z^{\Delta }\left(\theta \right)\right|\Delta \theta \ge \sqrt{q\left(t\right)}{\int }_{\alpha }^t\left|z^{\Delta }\left(\theta \right)\right|\Delta \theta \ge \sqrt{q\left(t\right)}z\left(t\right),$$

it follows that

$$2{\int }_{\alpha }^{\Gamma }q\left(\theta \right)|x\left(\theta \right)|\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta \le {\left({\int }_{\alpha }^{\Gamma }\frac{1}{\sqrt{p\left(\theta \right)}}\sqrt{p\left(\theta \right)q\left(\theta \right)}\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta \right)}^2.$$

By applying the Cauchy–Schwarz inequality, we get that

$${\int }_{\alpha }^{\Gamma }q\left(\theta \right)|x\left(\theta \right)|\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta \le \frac{1}{2}{\int }_{\alpha }^{\Gamma }\frac{\Delta \theta }{p\left(\theta \right)}{\int }_{\alpha }^{\Gamma }p\left(\theta \right)q\left(\theta \right){\left|x^{\Delta }\left(\theta \right)\right|}^2\Delta \theta .$$

(22)

Remark 5.

When $\mathbb{T}=\mathbb{R}$, the inequality (22), reduces to Yang inequality [57]

$${\int }_{\alpha }^{\Gamma }q\left(\theta \right)|x\left(\theta \right)|\left|x^{\prime }\left(\theta \right)\right|\mathrm{d}\theta \le \frac{1}{2}{\int }_{\alpha }^{\Gamma }\frac{\mathrm{d}\theta }{p\left(\theta \right)}{\int }_{\alpha }^{\Gamma }p\left(\theta \right)q\left(\theta \right){\left|x^{\prime }\left(\theta \right)\right|}^2\mathrm{d}\theta .$$

Corollary 8.

Take $f\left(t\right)=t^2$, $g\left(t\right)=1$, $m=1$ and $x=y$ in Theorem 5, the inequality (10) becomes

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }|x\left(\theta \right)||x^{\Delta }\left(\theta \right)|\Delta \theta \le & \frac{1}{2}{({\int }_{\alpha }^{\Gamma }|x^{\Delta }\left(\theta \right)|\Delta \theta )}^2\hfill \\ \hfill =& \frac{1}{2}{\left({\int }_{\alpha }^{\Gamma }{p}^{\frac{1}{\widehat{\nu }}}\left(\theta \right)p^{-\frac{1}{\widehat{\nu }}}\left(\theta \right)\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta \right)}^2.\hfill \end{array}$$

From Lemma 1 with indices $\widehat{\nu }$, $\widehat{\mu }$ such that $\frac{1}{\widehat{\nu }}+\frac{1}{\widehat{\mu }}=1$, we obtain

$${\int }_{\alpha }^{\Gamma }|x\left(\theta \right)||x^{\Delta }\left(\theta \right)|\Delta \theta \le \frac{1}{2}{\left({\int }_{\alpha }^{\Gamma }{p}^{1-\widehat{\mu }}\left(\theta \right)\Delta \theta \right)}^{\frac{2}{\widehat{\mu }}}{({\int }_{\alpha }^{\Gamma }p\left(\theta \right)|x^{\Delta }\left(\theta \right)|{}^{\widehat{\nu }}\Delta \theta )}^{\frac{2}{\widehat{\nu }}}.$$

(23)

Remark 6.

If $\mathbb{T}=\mathbb{R}$, then (23), gives the inequality of Maroni [58].

$${\int }_{\alpha }^{\Gamma }|x\left(\theta \right)||x^{\Delta }\left(\theta \right)|\mathrm{d}\theta \le \frac{1}{2}{\left({\int }_{\alpha }^{\Gamma }{p}^{1-\widehat{\mu }}\left(\theta \right)\mathrm{d}\theta \right)}^{\frac{2}{\widehat{\mu }}}{\left({\int }_{\alpha }^{\Gamma }p\left(\theta \right){|x^{\prime }\left(\theta \right)|}^{\nu }\mathrm{d}\theta \right)}^{\frac{2}{\nu }}.$$

(24)

Theorem 6.

Under the hypotheses of Theorem 5. Let $p^{\ast }>0$ and $p>0$ be defined on ${[\alpha ,\Gamma ]}_{\mathbb{T}}$ with ${\int }_{\alpha }^{\Gamma }p\left(\theta \right)\Delta \theta =1$ and ${\int }_{\alpha }^{\Gamma }{p}^{\ast }\left(\theta \right)\Delta \theta =1$. Further, let $\eta >0$ be convex and increasing on ${[0,\infty )}_{\mathbb{R}}$. Then, for all $m\ge 1$, the following inequality holds

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& \left[f\left({|x\left(\theta \right)|}^m\right)g^{\prime }\left({|y\left(\theta \right)|}^m\right){\left|y^{\Delta }\left(\theta \right)\right|}^m+g\left({|y\left(\theta \right)|}^m\right)f^{\prime }\left({|x\left(\theta \right)|}^m\right){\left|x^{\Delta }\left(\theta \right)\right|}^m\right]\Delta \theta \hfill \\ \hfill \le & \frac{1}{{\lambda }_1}f\left(2{\lambda }_1{\eta }^{-1}\left({\int }_{\alpha }^{\Gamma }p\left(\theta \right)\eta \left(\frac{|x^{\Delta }{\left(\theta \right)|}^m}{2p\left(\theta \right)}\right)\Delta \theta \right)\right)\hfill \\ \hfill & \times g\left(2{\lambda }_1{\eta }^{-1}\left({\int }_{\alpha }^{\Gamma }{p}^{\ast }\left(\theta \right)\eta \left(\frac{|y^{\Delta }{\left(\theta \right)|}^m}{2p^{\ast }\left(\theta \right)}\right)\Delta \theta \right)\right),\hfill \end{array}$$

(25)

where ${\lambda }_1$ defined as in Theorem 5.

Proof. 

Dynamic Jensen inequality (2) provides

$$\eta \left(\frac{1}{2}{\int }_{\alpha }^{\Gamma }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)\le {\int }_{\alpha }^{\Gamma }p\left(\theta \right)\eta \left(\frac{|x^{\Delta }{\left(\theta \right)|}^m}{2p\left(\theta \right)}\right)\Delta \theta .$$

Further, since $\eta$ is increasing, we have

$${\int }_{\alpha }^{\Gamma }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \le 2{\eta }^{-1}\left({\int }_{\alpha }^{\Gamma }p\left(\theta \right)\eta \left(\frac{|x^{\Delta }{\left(\theta \right)|}^m}{2p\left(\theta \right)}\right)\Delta \theta \right).$$

(26)

Similarly, we have

$${\int }_{\alpha }^{\Gamma }{|y^{\Delta }\left(\theta \right)|}^m\Delta \theta \le 2{\eta }^{-1}\left({\int }_{\alpha }^{\Gamma }{p}^{\ast }\left(\theta \right)\eta \left(\frac{|y^{\Delta }{\left(\theta \right)|}^m}{2p^{\ast }\left(\theta \right)}\right)\Delta \theta \right).$$

(27)

From (10), (26) and (27) immediately gives

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& \left[f\left({|x\left(\theta \right)|}^m\right)g^{\prime }\left({|y\left(\theta \right)|}^m\right){\left|y^{\Delta }\left(\theta \right)\right|}^m+g\left({|y\left(\theta \right)|}^m\right)f^{\prime }\left({|x\left(\theta \right)|}^m\right){\left|x^{\Delta }\left(\theta \right)\right|}^m\right]\Delta \theta \hfill \\ \hfill \le & \frac{1}{{\lambda }_1}f\left(2{\lambda }_1{\eta }^{-1}\left({\int }_{\alpha }^{\Gamma }p\left(\theta \right)\eta \left(\frac{|x^{\Delta }{\left(\theta \right)|}^m}{2p\left(\theta \right)}\right)\Delta \theta \right)\right)\hfill \\ \hfill & \times g\left(2{\lambda }_1{\eta }^{-1}\left({\int }_{\alpha }^{\Gamma }{p}^{\ast }\left(\theta \right)\eta \left(\frac{|y^{\Delta }{\left(\theta \right)|}^m}{2p^{\ast }\left(\theta \right)}\right)\Delta \theta \right)\right).\hfill \end{array}$$

This gives our claim. □

Remark 7.

When $\mathbb{T}=\mathbb{R}$ in Theorem 6, then, by the relation (4), we get the inequality of Hwang and Yang [3].

Corollary 9.

If we take $\mathbb{T}=h\mathbb{Z}$ in Theorem 6, then, by the relation (6), inequality (25) becomes

$$\begin{array}{cc}\hfill h\sum _{n=\frac{\alpha }{h}}^{\frac{\Gamma }{h}-1}& \left[f\left({|x\left(nh\right)|}^m\right)g^{\prime }\left({|y\left(nh\right)|}^m\right){\left|\Delta y\left(nh\right)\right|}^m+g\left({|y\left(nh\right)|}^m\right)f^{\prime }\left({|x\left(nh\right)|}^m\right){\left|\Delta x\left(nh\right)\right|}^m\right]\hfill \\ \hfill \le & \frac{1}{{\lambda }_1}f\left(2h{\lambda }_1{\eta }^{-1}\left(\sum _{n=\frac{\alpha }{h}}^{\frac{\Gamma }{h}-1}p\left(nh\right)\eta \left(\frac{{|\Delta x\left(nh\right)|}^m}{2p\left(nh\right)}\right)\right)\right)\hfill \\ \hfill & \times g\left(2h{\lambda }_1{\eta }^{-1}\left(\sum _{n=\frac{\alpha }{h}}^{\frac{\Gamma }{h}-1}{p}^{\ast }\left(nh\right)\eta \left(\frac{{|\Delta y\left(nh\right)|}^m}{2p^{\ast }\left(nh\right)}\right)\right)\right).\hfill \end{array}$$

Remark 8.

In Corollary 10, if we take $h=1$, then inequality (25) becomes

$$\begin{array}{cc}\hfill \sum _{n=\alpha }^{\Gamma -1}& \left[f\left({|x\left(n\right)|}^m\right)g^{\prime }\left({|y\left(n\right)|}^m\right){\left|\Delta y\left(n\right)\right|}^m+g\left({|y\left(n\right)|}^m\right)f^{\prime }\left({|x\left(n\right)|}^m\right){\left|\Delta x\left(n\right)\right|}^m\right]\hfill \\ \hfill \le & \frac{1}{{\lambda }_1}f\left(2{\lambda }_1{\eta }^{-1}\left(\sum _{n=\alpha }^{\Gamma -1}p\left(n\right)\eta \left(\frac{{|\Delta x\left(n\right)|}^m}{2p\left(n\right)}\right)\right)\right)\hfill \\ \hfill & \times g\left(2{\lambda }_1{\eta }^{-1}\left(\sum _{n=\alpha }^{\Gamma -1}{p}^{\ast }\left(n\right)\eta \left(\frac{{|\Delta y\left(n\right)|}^m}{2p^{\ast }\left(n\right)}\right)\right)\right).\hfill \end{array}$$

Corollary 10.

If we take $\mathbb{T}=\overline{q^{\mathbb{Z}}}$ in Theorem 6, then, by the relation (7), inequality (25) becomes

$$\begin{array}{cc}\hfill (q-1)\sum _{n=\left({\log }_q\alpha \right)}^{({\log }_q\Gamma )-1}& \left[f\left(|x\left(q^n\right){|}^m\right)g^{\prime }\left(|y\left(q^n\right){|}^m\right){\left|\Delta y\left(q^n\right)\right|}^m+g\left(|y\left(q^n\right){|}^m\right)f^{\prime }\left(|x\left(q^n\right){|}^m\right){\left|\Delta x\left(q^n\right)\right|}^m\right]q^n\hfill \\ \hfill \le & \frac{1}{{\lambda }_1}f\left(2(q-1){\lambda }_1{\eta }^{-1}\left(\sum _{n=\left({\log }_q\alpha \right)}^{({\log }_q\Gamma )-1}{q}^{n}p\left(q^n\right)\eta \left(\frac{|\Delta x\left(q^n\right){|}^m}{2p\left(q^n\right)}\right)\right)\right)\hfill \\ \hfill & \times g\left(2(q-1){\lambda }_1{\eta }^{-1}\left(\sum _{n=\left({\log }_q\alpha \right)}^{({\log }_q\Gamma )-1}{q}^n{p}^{\ast }\left(q^n\right)\eta \left(\frac{|\Delta y\left(q^n\right){|}^m}{2p^{\ast }\left(q^n\right)}\right)\right)\right).\hfill \end{array}$$

Remark 9.

When $\mathbb{T}=\mathbb{R}$, if we take $m=1$ and $p\left(t\right)=p^{\ast }\left(t\right)$, in the inequality (25), we get Pachpatte inequality [54]

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& \left[f(|x\left(\theta \right)|)g^{\prime }(|y\left(\theta \right)|)|y^{\prime }\left(\theta \right)|+g(|y\left(\theta \right)|)f^{\prime }(|x\left(\theta \right)|)|x^{\prime }\left(\theta \right)|\right]\mathrm{d}\theta \hfill \\ \hfill \le & f\left({\eta }^{-1}\left({\int }_{\alpha }^{\Gamma }{p}^{\ast }\left(\theta \right)\eta \left(\frac{|x^{\prime }\left(\theta \right)|}{p^{\ast }\left(\theta \right)}\right)\mathrm{d}\theta \right)\right)\hfill \\ \hfill & \times g\left({\eta }^{-1}\left({\int }_{\alpha }^{\Gamma }{p}^{\ast }\left(\theta \right)\eta \left(\frac{|y^{\prime }\left(\theta \right)|}{p^{\ast }\left(\theta \right)}\right)\mathrm{d}\theta \right)\right).\hfill \end{array}$$

(28)

Remark 10.

Taking $f\left(\theta \right)=g\left(\theta \right)$ and $x\left(\theta \right)=y\left(\theta \right)$ in inequality (28), we get the Pachpatte inequality [54]

$$\begin{array}{c}\hfill {\int }_{\alpha }^{\Gamma }f(|x\left(\theta \right)|)f^{\prime }(|x\left(\theta \right)|)|x^{\prime }\left(\theta \right)|\mathrm{d}\theta \le \frac{1}{2}{\left[f\left({\eta }^{-1}\left({\int }_{\alpha }^{\Gamma }p\left(\theta \right)\eta \left(\frac{|x^{\prime }\left(\theta \right)|}{p\left(\theta \right)}\right)\mathrm{d}\theta \right)\right)\right]}^2.\end{array}$$

Corollary 11.

Suppose $\mathbb{T}$ is a time scale, α, β, $\raisebox{1ex}{$(\alpha +\beta )$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\in \mathbb{T}$, and f, g are defined as in Theorem 5. Furthermore, Assume x and y are rd-continuous functions on ${[\alpha ,\beta ]}_{\mathbb{T}}$ such that $x\left(\alpha \right)=y\left(\alpha \right)=0$ and $x\left(\beta \right)=y\left(\beta \right)=0$. Then, we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\beta }& \left[{f(|x\left(\theta \right)|}^m)g^{\prime }{(|y\left(\theta \right)|}^m)|y^{\Delta }{\left(\theta \right)|}^m+{g(|y\left(\theta \right)|}^m)f^{\prime }{(|x\left(\theta \right)|}^m)|x^{\Delta }{\left(\theta \right)|}^m\right]\Delta \theta \hfill \\ \hfill \le & \frac{2^{m-1}}{{\lambda }_2}g\left(2^{1-m}{\lambda }_2{\int }_{\alpha }^{\beta }{|y^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)\hfill \\ \hfill & \times \left[f\left(2^{1-m}{\lambda }_2{\int }_{\alpha }^{\frac{\alpha +\beta }{2}}{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)+f\left(2^{1-m}{\lambda }_2{\int }_{\frac{\alpha +\beta }{2}}^{\beta }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)\right],\hfill \end{array}$$

(29)

where ${\lambda }_2={(\beta -\alpha )}^{m-1}$.

Proof. 

Let $\Gamma \in {[\alpha ,\beta ]}_{\mathbb{T}}$, the functions x and y satisfy the conditions of Theorem 5 on ${[\alpha ,\Gamma ]}_{\mathbb{T}}$. Thus, inequality (10) holds. Next, in the interval ${[\Gamma ,\beta ]}_{\mathbb{T}}$, the functions x and y are rd-continuous, and $x\left(\beta \right)=y\left(\beta \right)=0$. Thus, by defining $\phi \left(t\right)={\int }_t^{\beta }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta$, $\varphi \left(t\right)={\int }_t^{\beta }{|y^{\Delta }\left(\theta \right)|}^m\Delta \theta$$t\in [\Gamma ,\beta ]$, and following an argument similar to Theorem 5, we obtain

$$\begin{array}{cc}\hfill {\int }_{\Gamma }^{\beta }& \left[{f(|x\left(\theta \right)|}^m)g^{\prime }{(|y\left(\theta \right)|}^m)|y^{\Delta }{\left(\theta \right)|}^m+{g(|y\left(\theta \right)|}^m)f^{\prime }{(|x\left(\theta \right)|}^m)|x^{\Delta }{\left(\theta \right)|}^m\right]\Delta \theta \hfill \\ \hfill \le & \frac{1}{{\lambda }_3}f\left({\lambda }_3{\int }_{\Gamma }^{\beta }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)g\left({\lambda }_3{\int }_{\Gamma }^{\beta }{|y^{\Delta }\left(\theta \right)|}^m\Delta \theta \right),\hfill \end{array}$$

(30)

where ${\lambda }_3={(\beta -\Gamma )}^{m-1}$. A combination of the inequalities (10) and (30), we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\beta }& \left[{f(|x\left(\theta \right)|}^m)g^{\prime }{(|y\left(\theta \right)|}^m)|y^{\Delta }{\left(\theta \right)|}^m+{g(|y\left(\theta \right)|}^m)f^{\prime }{(|x\left(\theta \right)|}^m)|x^{\Delta }{\left(\theta \right)|}^m\right]\Delta \theta \hfill \\ \hfill =& {\int }_{\alpha }^{\Gamma }\left[{f(|x\left(\theta \right)|}^m)g^{\prime }{(|y\left(\theta \right)|}^m)|y^{\Delta }{\left(\theta \right)|}^m+{g(|y\left(\theta \right)|}^m)f^{\prime }{(|x\left(\theta \right)|}^m)|x^{\Delta }{\left(\theta \right)|}^m\right]\Delta \theta \hfill \\ \hfill & +{\int }_{\Gamma }^{\beta }\left[{f(|x\left(\theta \right)|}^m)g^{\prime }{(|y\left(t\right)|}^m)|y^{\Delta }{\left(\theta \right)|}^m+{g(|y\left(\theta \right)|}^m)f^{\prime }{(|x\left(\theta \right)|}^m)|x^{\Delta }{\left(\theta \right)|}^m\right]\Delta \theta \hfill \\ \hfill \le & \frac{1}{{\lambda }_1}f\left({\lambda }_1{\int }_{\alpha }^{\Gamma }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)g\left({\lambda }_1{\int }_{\alpha }^{\Gamma }{y}^{\Delta }{\left(\theta \right)|}^m\Delta \theta \right)\hfill \\ \hfill & +\frac{1}{{\lambda }_3}f\left({\lambda }_3{\int }_{\Gamma }^{\beta }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)g\left({\lambda }_3{\int }_{\Gamma }^{\beta }{|y^{\Delta }\left(\theta \right)|}^m\Delta \theta \right),\hfill \end{array}$$

(31)

for $\Gamma =\frac{\alpha +\beta }{2}$, we find that ${\lambda }_1={\lambda }_3={\left(\frac{\beta -\alpha }{2}\right)}^{m-1}$, where ${\lambda }_2={(\beta -\alpha )}^{m-1}$, then ${\lambda }_1={\lambda }_3=\frac{{\lambda }_2}{2^{m-1}}$. Then, by substituting in (31), we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\beta }& \left[{f(|x\left(\theta \right)|}^m)g^{\prime }{(|y\left(\theta \right)|}^m)|y^{\Delta }{\left(\theta \right)|}^m+{g(|y\left(\theta \right)|}^m)f^{\prime }{(|x\left(\theta \right)|}^m)|x^{\Delta }{\left(\theta \right)|}^m\right]\Delta \theta \hfill \\ \hfill \le & \frac{2^{m-1}}{{\lambda }_2}f\left(2^{1-m}{\lambda }_2{\int }_{\alpha }^{\frac{\alpha +\beta }{2}}{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)g\left(2^{1-m}{\lambda }_2{\int }_{\alpha }^{\frac{\alpha +\beta }{2}}{|y^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)\hfill \\ \hfill & +\frac{2^{m-1}}{{\lambda }_2}f\left(2^{1-m}{\lambda }_2{\int }_{\frac{\alpha +\beta }{2}}^{\beta }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)g\left(2^{1-m}{\lambda }_2{\int }_{\frac{\alpha +\beta }{2}}^{\beta }{|y^{\Delta }\left(\theta \right)|}^m\Delta \theta \right).\hfill \end{array}$$

(32)

Since g is non-decreasing function, we have

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\beta }& \left[{f(|x\left(\theta \right)|}^m)g^{\prime }{(|y\left(\theta \right)|}^m)|y^{\Delta }{\left(\theta \right)|}^m+{g(|y\left(\theta \right)|}^m)f^{\prime }{(|x\left(t\right)|}^m)|x^{\Delta }{\left(\theta \right)|}^m\right]\Delta \theta \hfill \\ \hfill \le & \frac{2^{m-1}}{{\lambda }_2}g\left(2^{1-m}{\lambda }_2{\int }_{\alpha }^{\beta }{|y^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)[f\left(2^{1-m}{\lambda }_2{\int }_{\alpha }^{\frac{\alpha +\beta }{2}}{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)\hfill \\ \hfill & +f\left(2^{1-m}{\lambda }_2{\int }_{\frac{\alpha +\beta }{2}}^{\beta }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)].\hfill \end{array}$$

This gives our claim (29). □

Corollary 12.

If we take $\mathbb{T}=h\mathbb{Z}$ in Corollary 11, then, by the relation (6), inequality (29) becomes

$$\begin{array}{cc}\hfill h\sum _{n=\frac{\alpha }{h}}^{\frac{\beta }{h}-1}& \left[{f(|x\left(nh\right)|}^m)g^{\prime }{(|y\left(nh\right)|}^m{)|\Delta y\left(nh\right)|}^m+{g(|y\left(nh\right)|}^m)f^{\prime }{(|x\left(nh\right)|}^m{)|\Delta x\left(nh\right)|}^m\right]\hfill \\ \hfill \le & \frac{2^{m-1}}{{\lambda }_2}g\left(2^{1-m}{\lambda }_{2}h\sum _{n=\frac{\alpha }{h}}^{\frac{\beta }{h}-1}{|\Delta y\left(nh\right)|}^m\right)\hfill \\ \hfill & \times \left[f\left(2^{1-m}{\lambda }_{2}h\sum _{n=\frac{\alpha }{h}}^{\frac{\alpha +\beta }{2h}-1}{|\Delta x\left(nh\right)|}^m\right)+f\left(2^{1-m}{\lambda }_{2}h\sum _{n=\frac{\alpha +\beta }{2h}}^{\frac{\beta }{h}-1}{|\Delta x\left(nh\right)|}^m\right)\right].\hfill \end{array}$$

Remark 11.

In Corollary 12, if we take $h=1$, then inequality (29) becomes

$$\begin{array}{cc}\hfill \sum _{n=\alpha }^{\beta -1}& \left[{f(|x\left(n\right)|}^m)g^{\prime }{(|y\left(n\right)|}^m{)|\Delta y\left(n\right)|}^m+{g(|y\left(n\right)|}^m)f^{\prime }{(|x\left(n\right)|}^m{)|\Delta x\left(n\right)|}^m\right]\hfill \\ \hfill \le & \frac{2^{m-1}}{{\lambda }_2}g\left(2^{1-m}{\lambda }_2\sum _{n=\alpha }^{\beta -1}{|\Delta y\left(n\right)|}^m\right)\hfill \\ \hfill & \times \left[f\left(2^{1-m}{\lambda }_2\sum _{n=\alpha }^{\frac{\alpha +\beta }{2}-1}{|\Delta x\left(n\right)|}^m\right)+f\left(2^{1-m}{\lambda }_2\sum _{n=\frac{\alpha +\beta }{2}}^{\beta -1}{|\Delta x\left(n\right)|}^m\right)\right].\hfill \end{array}$$

Corollary 13.

If we take $\mathbb{T}=\overline{q^{\mathbb{Z}}}$ in Corollary 11, then, by the relation (7), inequality (29) becomes

$$\begin{array}{cc}\hfill (q-1)\sum _{n=\left({\log }_q\alpha \right)}^{\left({\log }_q\beta \right)-1}& \left[f(|x\left(q^n\right){|}^m)g^{\prime }(|y\left(q^n\right){|}^m)|\Delta y\left(q^n\right){|}^m+g(|y\left(q^n\right){|}^m)f^{\prime }(|x\left(q^n\right){|}^m)|\Delta x\left(q^n\right){|}^m\right]q^n\hfill \\ \hfill \le & \frac{2^{m-1}}{{\lambda }_2}g\left(2^{1-m}{\lambda }_2(q-1)\sum _{n=\left({\log }_q\alpha \right)}^{\left({\log }_q\beta \right)-1}{|\Delta y\left(q^n\right)|}^m{q}^n\right)\hfill \\ \hfill & \times [f\left(2^{1-m}{\lambda }_2(q-1)\sum _{n=\left({\log }_q\alpha \right)}^{\left({\log }_q\frac{\alpha +\beta }{2}\right)-1}{|\Delta x\left(q^n\right)|}^m{q}^n\right)\hfill \\ \hfill & +f\left(2^{1-m}{\lambda }_2(q-1)\sum _{n=\left({\log }_q\frac{\alpha +\beta }{2}\right)}^{\left({\log }_q\beta \right)-1}{|\Delta x\left(q^n\right)|}^m{q}^n\right)].\hfill \end{array}$$

Corollary 14.

In Corollary 11. For $m=1$, we can get the following inequality

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\beta }& \left[f(|x\left(\theta \right)|)g^{\prime }(|y\left(\theta \right)|)|y^{\Delta }\left(\theta \right)|+g(|y\left(t\right)|)f^{\prime }(|x\left(\theta \right)|)|x^{\Delta }\theta |\right]\Delta \theta \hfill \\ \hfill & \le 2f\left(\frac{1}{2}{\int }_{\alpha }^{\beta }|x^{\Delta }\left(\theta \right)|\Delta \theta \right)g\left(\frac{1}{2}{\int }_{\alpha }^{\beta }|y^{\Delta }\left(\theta \right)|\Delta \theta \right).\hfill \end{array}$$

(33)

Proof. 

By substituting in (32) by $m=1$, we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\beta }& \left[f(|x\left(\theta \right)|)g^{\prime }(|y\left(\theta \right)|)|y^{\Delta }\left(\theta \right)|+g(|y\left(\theta \right)|)f^{\prime }(|x\left(\theta \right)|)|x^{\Delta }\left(\theta \right)|\right]\Delta \theta \hfill \\ \hfill \le & f\left({\int }_{\alpha }^{\frac{\alpha +\beta }{2}}|x^{\Delta }\left(\theta \right)|\Delta \theta \right)g\left({\int }_{\alpha }^{\frac{\alpha +\beta }{2}}|y^{\Delta }\left(\theta \right)|\Delta \theta \right)\hfill \\ \hfill & +f\left({\int }_{\frac{\alpha +\beta }{2}}^{\beta }|x^{\Delta }\left(\theta \right)|\Delta \theta \right)g\left({\int }_{\frac{\alpha +\beta }{2}}^{\beta }|y^{\Delta }\left(\theta \right)|\Delta \theta \right).\hfill \end{array}$$

(34)

If we choose $\Gamma$ such that

$${\int }_{\alpha }^{\Gamma }\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta ={\int }_{\Gamma }^{\beta }\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta =\frac{1}{2}{\int }_{\alpha }^{\beta }\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta .$$

(35)

Then, by substituting from (35) in (34), we can get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\beta }& \left[f(|x\left(\theta \right)|)g^{\prime }(|y\left(\theta \right)|)|y^{\Delta }\left(\theta \right)|+g(|y\left(\theta \right)|)f^{\prime }(|x\left(\theta \right)|)|x^{\Delta }\left(\theta \right)|\right]\Delta \theta \hfill \\ \hfill \le & 2f\left(\frac{1}{2}{\int }_{\alpha }^{\beta }|x^{\Delta }\left(\theta \right)|\Delta \theta \right)g\left(\frac{1}{2}{\int }_{\alpha }^{\beta }|y^{\Delta }\left(\theta \right)|\Delta \theta \right).\hfill \end{array}$$

This gives our claim. □

Corollary 15.

Let $f\left(t\right)=t^{\frac{\ell +m}{m}}$, $\ell \ge 0$, $g\left(t\right)=1$ and $x\left(t\right)=y\left(t\right)$ in (29), to obtain

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\beta }\frac{\ell +m}{m}{|x\left(\theta \right)|}^{\ell }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \le & {\left(\frac{\beta -\alpha }{2}\right)}^{\frac{\ell (m-1)}{m}}[{\left({\int }_{\alpha }^{\frac{\alpha +\beta }{2}}{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)}^{\frac{\ell +m}{m}}\hfill \\ \hfill & +{\left({\int }_{\frac{\alpha +\beta }{2}}^{\beta }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)}^{\frac{\ell +m}{m}}].\hfill \end{array}$$

(36)

By using the inequality (9) in inequality (36), we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\beta }{|x\left(\theta \right)|}^{\ell }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \le & \frac{m}{\ell +m}{\left(\frac{\beta -\alpha }{2}\right)}^{\frac{\ell (m-1)}{m}}\hfill \\ \hfill & \times \left[{\left({\int }_{\alpha }^{\frac{\alpha +\beta }{2}}{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)}^{\frac{\ell +m}{m}}+{\left({\int }_{\frac{\alpha +\beta }{2}}^{\beta }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)}^{\frac{\ell +m}{m}}\right]\hfill \\ \hfill \le & \frac{m}{\ell +m}{\left(\frac{\beta -\alpha }{2}\right)}^{\frac{\ell (m-1)}{m}}{\left[\left({\int }_{\alpha }^{\frac{\alpha +\beta }{2}}{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)+\left({\int }_{\frac{\alpha +\beta }{2}}^{\beta }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)\right]}^{\frac{\ell +m}{m}}\hfill \\ \hfill =& \frac{m}{\ell +m}{\left(\frac{\beta -\alpha }{2}\right)}^{\frac{\ell (m-1)}{m}}{\left({\int }_{\alpha }^{\beta }{|x^{\Delta }\left(\theta \right)|}^m\Delta \theta \right)}^{\frac{\ell +m}{m}}.\hfill \end{array}$$

(37)

From Lemma 1 with indices $\frac{\ell +m}{\ell }$ and $\frac{\ell +m}{m}$ on (37), we have

$${\int }_{\alpha }^{\beta }{|x\left(\theta \right)|}^{\ell }|x^{\Delta }{\left(\theta \right)|}^m\Delta \theta \le \frac{m}{\ell +m}{\left(\frac{\beta -\alpha }{2}\right)}^{\ell }{\int }_{\alpha }^{\beta }{|x^{\Delta }\left(\theta \right)|}^{\ell +m}\Delta \theta .$$

(38)

Remark 12.

Taking $\mathbb{T}=\mathbb{R}$, then (38) gives Yang inequality [59]

$${\int }_{\alpha }^{\beta }{|x\left(\theta \right)|}^{\ell }|x^{\prime }{\left(\theta \right)|}^m\mathrm{d}t\le \frac{m}{\ell +m}{\left(\frac{\beta -\alpha }{2}\right)}^{\ell }{\int }_{\alpha }^{\beta }{|x^{\prime }\left(\theta \right)|}^{\ell +m}\mathrm{d}\theta .$$

Theorem 7.

Under the hypotheses of Theorem 5. Assuming that $s\ge 0$, $r\ge 0$ and $s^{\Delta }\ge 0$, $r^{\Delta }\ge 0$, $t\in {[\alpha ,\Gamma ]}_{\mathbb{T}}$ and $r\left(\alpha \right)=0$, $s\left(\alpha \right)=0$ and $\chi \ge 0$ and $\pi \ge 0$ are convex and increasing functions on ${(0,\infty )}_{\mathbb{R}}$. Then, we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& [s^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)f\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g^{\prime }\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)\hfill \\ \hfill & +r^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)f^{\prime }\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)]\Delta \theta \hfill \\ \hfill \le & f\left({\int }_{\alpha }^{\Gamma }{r}^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)g\left({\int }_{\alpha }^{\Gamma }{s}^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right).\hfill \end{array}$$

(39)

Proof. 

For $t\in {[\alpha ,\Gamma ]}_{\mathbb{T}}$, we define $\phi \left(t\right):={\int }_{\alpha }^t\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta$ and $\psi \left(t\right):={\int }_{\alpha }^t\left|y^{\Delta }\left(\theta \right)\right|\Delta \theta$ so that ${\phi }^{\Delta }\left(t\right)=\left|x^{\Delta }\left(t\right)\right|$ and ${\psi }^{\Delta }\left(t\right)=\left|y^{\Delta }\left(t\right)\right|$.

Thus,

$$\begin{array}{cc}\hfill |x(t)|=& \left|{\int }_{\alpha }^t{x}^{\Delta }\left(\theta \right)\Delta \theta \right|\le {\int }_{\alpha }^t\left|x^{\Delta }\left(\theta \right)\right|\Delta \theta =\phi \left(t\right),\hfill \\ \hfill |y(t)|=& \left|{\int }_{\alpha }^t{y}^{\Delta }\left(\theta \right)\Delta \theta \right|\le {\int }_{\alpha }^t\left|y^{\Delta }\left(\theta \right)\right|\Delta \theta =\psi \left(t\right).\hfill \end{array}$$

Then, we obtain

$$\frac{|x(t)|}{|r(t)|}\le \frac{\frac{{\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right){\phi }^{\Delta }\left(\theta \right)\Delta \theta }{r^{\Delta }\left(\theta \right)}}{{\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\Delta \theta },$$

$$\frac{|y(t)|}{|s(t)|}\le \frac{\frac{{\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right){\psi }^{\Delta }\left(\theta \right)\Delta \theta }{s^{\Delta }\left(\theta \right)}}{{\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\Delta \theta }.$$

Thus, from Lemma 2, we get

$$\chi \left(\frac{|x(t)|}{r\left(t\right)}\right)\le \frac{1}{r\left(t\right)}{\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta .$$

$$\pi \left(\frac{|y(t)|}{s\left(t\right)}\right)\le \frac{1}{s\left(t\right)}{\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta .$$

Using the above inequalities, we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& [s^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)f\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g^{\prime }\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)\hfill \\ \hfill & +r^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)f^{\prime }\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)]\Delta \theta \hfill \\ \hfill \le & {\int }_{\alpha }^{\Gamma }[s^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)f\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\Delta \theta \right)g^{\prime }\left({\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)\hfill \\ \hfill & +r^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)f^{\prime }\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)g\left({\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)]\Delta \theta .\hfill \end{array}$$

(40)

Since f, r, $\chi$, and $\phi$ are increasing and $t\le \sigma \left(t\right)$, we get

$$f\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)\le f^{\sigma }\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta .}\left(\theta \right)}\right)\Delta \theta \right).$$

Then by substituting in (40), we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& [s^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)f\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g^{\prime }\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)\hfill \\ \hfill & +r^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)f^{\prime }\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)]\Delta \theta \hfill \\ \hfill & \le {\int }_{\alpha }^{\Gamma }[s^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)f^{\sigma }\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)g^{\prime }\left({\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)\hfill \\ \hfill & +r^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)f^{\prime }\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)g\left({\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)]\Delta \theta .\hfill \end{array}$$

(41)

From Lemma 3 for $c\in [t,\sigma (t\left)\right]$, we get

$$\begin{array}{cc}\hfill f^{\Delta }\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\Delta \theta \right)\right)=& r^{\Delta }\left(t\right)\chi \left(\frac{{\phi }^{\Delta }\left(t\right)}{r^{\Delta }\left(t\right)}\right)f^{\prime }\left({\int }_{\alpha }^c{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)\hfill \\ \hfill \ge & r^{\Delta }\left(t\right)\chi \left(\frac{{\phi }^{\Delta }\left(t\right)}{r^{\Delta }\left(t\right)}\right)f^{\prime }\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right).\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill g^{\Delta }\left({\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\Delta \theta \right)\right)=& s^{\Delta }\left(t\right)\chi \left(\frac{{\psi }^{\Delta }\left(t\right)}{s^{\Delta }\left(t\right)}\right)g^{\prime }\left({\int }_{\alpha }^c{s}^{\Delta }\left(\theta \right)\chi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)\hfill \\ \hfill \ge & s^{\Delta }\left(t\right)\chi \left(\frac{{\phi }^{\Delta }\left(t\right)}{s^{\Delta }\left(t\right)}\right)g^{\prime }\left({\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right).\hfill \end{array}$$

(43)

Then, by substituting from (42) and (42) in (41), we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& [s^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)f\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g^{\prime }\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)\hfill \\ \hfill & +r^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)f^{\prime }\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)]\Delta \theta \hfill \\ \hfill \le & {\int }_{\alpha }^{\Gamma }[g^{{}^{\Delta }}\left({\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)f^{\sigma }\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)\hfill \\ \hfill & +f^{\Delta }\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)g\left({\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)]\Delta \theta \hfill \\ \hfill =& {\int }_{\alpha }^{\Gamma }[f\left({\int }_{\alpha }^t{r}^{\Delta }\left(\theta \right)\chi \left(\frac{{\phi }^{\Delta }\left(\theta \right)}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)g\left({\int }_{\alpha }^t{s}^{\Delta }\left(\theta \right)\pi \left(\frac{{\psi }^{\Delta }\left(\theta \right)}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right){]}^{\Delta }\hfill \\ \hfill =& f\left({\int }_{\alpha }^{\Gamma }{r}^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right)g\left({\int }_{\alpha }^{\Gamma }{s}^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)\Delta \theta \right).\hfill \end{array}$$

This gives our claim. □

Remark 13.

Taking $\mathbb{T}=\mathbb{R}$, then, by the relation (6), inequality (39) gives Pachpatte inequality in [55]

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& [s^{\prime }\left(\theta \right)\pi \left(\frac{|y^{\prime }\left(\theta \right)|}{s^{\prime }\left(\theta \right)}\right)f\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g^{\prime }\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)\hfill \\ \hfill & +r^{\prime }\left(\theta \right)\chi \left(\frac{|x^{\prime }\left(\theta \right)|}{r^{\prime }\left(\theta \right)}\right)f^{\prime }\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)]\mathrm{d}\theta \hfill \\ \hfill \le & f\left({\int }_{\alpha }^{\Gamma }{r}^{\prime }\left(\theta \right)\chi \left(\frac{|x^{\prime }\left(\theta \right)|}{r^{\prime }\left(\theta \right)}\right)\mathrm{d}\theta \right)g\left({\int }_{\alpha }^{\Gamma }{s}^{\prime }\left(\theta \right)\pi \left(\frac{|y^{\prime }\left(\theta \right)|}{s^{\prime }\left(\theta \right)}\right)\mathrm{d}\theta \right).\hfill \end{array}$$

Corollary 16.

If we take $\mathbb{T}=h\mathbb{Z}$ in Theorem 7, then, by the relation (6), inequality (39) becomes

$$\begin{array}{cc}\hfill h\sum _{n=\frac{\alpha }{h}}^{\frac{\Gamma }{h}-1}& [\Delta s\left(nh\right)\pi \left(\frac{|\Delta y(nh)|}{\Delta s\left(nh\right)}\right)f\left(r\left(nh\right)\chi \left(\frac{|x(nh)|}{r\left(nh\right)}\right)\right)g^{\prime }\left(s\left(nh\right)\pi \left(\frac{|y(nh)|}{s\left(nh\right)}\right)\right)\hfill \\ \hfill & +\Delta r\left(nh\right)\chi \left(\frac{|\Delta x(nh)|}{\Delta r\left(nh\right)}\right)f^{\prime }\left(r\left(nh\right)\chi \left(\frac{|x(nh)|}{r\left(nh\right)}\right)\right)g\left(s\left(nh\right)\pi \left(\frac{|y(nh)|}{s\left(nh\right)}\right)\right)]\hfill \\ \hfill \le & f\left(h\sum _{n=\frac{\alpha }{h}}^{\frac{\Gamma }{h}-1}\Delta r\left(nh\right)\chi \left(\frac{|\Delta x(nh)|}{\Delta r\left(nh\right)}\right)\right)g\left(h\sum _{n=\frac{\alpha }{h}}^{\frac{\Gamma }{h}-1}\Delta s\left(nh\right)\pi \left(\frac{|\Delta y(nh)|}{\Delta s\left(nh\right)}\right)\right).\hfill \end{array}$$

Remark 14.

In Corollary 16, if we take $h=1$, then inequality (39) becomes

$$\begin{array}{cc}\hfill \sum _{n=\alpha }^{\Gamma -1}& [\Delta s\left(n\right)\pi \left(\frac{|\Delta y(n)|}{\Delta s\left(n\right)}\right)f\left(r\left(n\right)\chi \left(\frac{|x(n)|}{r\left(n\right)}\right)\right)g^{\prime }\left(s\left(n\right)\pi \left(\frac{|y(n)|}{s\left(n\right)}\right)\right)\hfill \\ \hfill & +\Delta r\left(n\right)\chi \left(\frac{|\Delta x(n)|}{\Delta r\left(n\right)}\right)f^{\prime }\left(r\left(n\right)\chi \left(\frac{|x(n)|}{r\left(n\right)}\right)\right)g\left(s\left(n\right)\pi \left(\frac{|y(n)|}{s\left(n\right)}\right)\right)]\hfill \\ \hfill \le & f\left(\sum _{n=\alpha }^{\Gamma -1}\Delta r\left(n\right)\chi \left(\frac{|\Delta x(n)|}{\Delta r\left(n\right)}\right)\right)g\left(\sum _{n=\alpha }^{\Gamma -1}\Delta s\left(n\right)\pi \left(\frac{|\Delta y(n)|}{\Delta s\left(n\right)}\right)\right).\hfill \end{array}$$

Corollary 17.

If we take $\mathbb{T}=\overline{q^{\mathbb{Z}}}$ in Theorem 7, then, by the relation (7), inequality (39) becomes

$$\begin{array}{cc}\hfill (q-1)\sum _{n=\left({\log }_q\alpha \right)}^{({\log }_q\Gamma )-1}& [\Delta s\left(q^n\right)\pi \left(\frac{|\Delta y(q^n)|}{\Delta s\left(q^n\right)}\right)f\left(r\left(q^n\right)\chi \left(\frac{|x(q^n)|}{r\left(q^n\right)}\right)\right)g^{\prime }\left(s\left(q^n\right)\pi \left(\frac{|y(q^n)|}{s\left(q^n\right)}\right)\right)\hfill \\ \hfill & +\Delta r\left(q^n\right)\chi \left(\frac{|\Delta x(q^n)|}{\Delta r\left(q^n\right)}\right)f^{\prime }\left(r\left(q^n\right)\chi \left(\frac{|x(q^n)|}{r\left(q^n\right)}\right)\right)g\left(s\left(q^n\right)\pi \left(\frac{|y(q^n)|}{s\left(q^n\right)}\right)\right)]q^n\hfill \\ \hfill \le & f\left((q-1)\sum _{n=\left({\log }_q\alpha \right)}^{({\log }_q\Gamma )-1}{q}^n\Delta r\left(q^n\right)\chi \left(\frac{|\Delta x(q^n)|}{\Delta r\left(q^n\right)}\right)\right)\hfill \\ \hfill & \times g\left((q-1)\sum _{n=\left({\log }_q\alpha \right)}^{({\log }_q\Gamma )-1}{q}^n\Delta s\left(q^n\right)\pi \left(\frac{|\Delta y(q^n)|}{\Delta s\left(q^n\right)}\right)\right).\hfill \end{array}$$

Remark 15.

For $s\left(t\right)=r\left(t\right)$, $\chi \left(t\right)=\pi \left(t\right)$, $y\left(t\right)=x\left(t\right)$ and $g\left(t\right)=f\left(t\right)$, the inequality (39) becomes

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& r^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)f\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)f^{\prime }\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)\Delta \theta \hfill \\ \hfill \le & \frac{1}{2}[f\left({\int }_{\alpha }^{\Gamma }{r}^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)\Delta \theta \right){]}^2.\hfill \end{array}$$

(44)

Corollary 18.

With the assumptions of Theorem 7. Suppose $\omega \ge 0$, $t\in {[\alpha ,\Gamma ]}_{\mathbb{T}}$ and ${\int }_{\alpha }^{\Gamma }\omega \left(\theta \right)\Delta \theta =1$. Assuming $\Psi \ge 0$ is increasing and convex on ${[0,\infty )}_{\mathbb{R}}$. Then, we get

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\Gamma }& [s^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)f\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g^{\prime }\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)\hfill \\ \hfill & +r^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)f^{\prime }\left(r\left(\theta \right)\chi \left(\frac{|x(\theta )|}{r\left(\theta \right)}\right)\right)g\left(s\left(\theta \right)\pi \left(\frac{|y(\theta )|}{s\left(\theta \right)}\right)\right)]\Delta \theta \hfill \\ \hfill \le & f\left({\Psi }^{-1}\left({\int }_{\alpha }^{\Gamma }\omega \left(\theta \right)\Psi \left(\frac{r^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)}{\omega \left(\theta \right)}\right)\Delta \theta \right)\right)g\left({\Psi }^{-1}\left({\int }_{\alpha }^{\Gamma }\omega \left(\theta \right)\Psi \left(\frac{s^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)}{\omega \left(\theta \right)}\right)\Delta \theta \right)\right).\hfill \end{array}$$

(45)

Proof. 

Since

$${\int }_{\alpha }^{\Gamma }{r}^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)\le \frac{{\int }_{\alpha }^{\Gamma }\frac{r^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)}{\omega \left(\theta \right)}\omega \left(\theta \right)\Delta \theta }{{\int }_{\alpha }^{\Gamma }\omega \left(\theta \right)\Delta \theta }$$

and

$${\int }_{\alpha }^{\Gamma }{s}^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)\le \frac{{\int }_{\alpha }^{\Gamma }\frac{s^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)}{\omega \left(\theta \right)}\omega \left(\theta \right)\Delta \theta }{{\int }_{\alpha }^{\Gamma }\omega \left(\theta \right)\Delta \theta }.$$

From Lemma 2, we have

$$\Psi \left({\int }_{\alpha }^{\Gamma }{r}^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)\right)\le {\int }_{\alpha }^{\Gamma }\omega \left(\theta \right)\Psi \left(\frac{r^{\Delta }\left(\theta \right)\pi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)}{\omega \left(\theta \right)}\right)\Delta \theta$$

and

$$\Psi \left({\int }_{\alpha }^{\Gamma }{s}^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)\right)\le {\int }_{\alpha }^{\Gamma }\omega \left(\theta \right)\Psi \left(\frac{s^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)}{\omega \left(\theta \right)}\right)\Delta \theta$$

and hence

$${\int }_{\alpha }^{\Gamma }{r}^{\Delta }\left(\theta \right)\chi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)\le {\Psi }^{-1}\left({\int }_{\alpha }^{\Gamma }\omega \left(\theta \right)\Psi \left(\frac{r^{\Delta }\left(\theta \right)\pi \left(\frac{|x^{\Delta }\left(\theta \right)|}{r^{\Delta }\left(\theta \right)}\right)}{\omega \left(\theta \right)}\right)\Delta \theta \right)$$

(46)

and

$${\int }_{\alpha }^{\Gamma }{s}^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)\le {\Psi }^{-1}\left({\int }_{\alpha }^{\Gamma }\omega \left(\theta \right)\Psi \left(\frac{s^{\Delta }\left(\theta \right)\pi \left(\frac{|y^{\Delta }\left(\theta \right)|}{s^{\Delta }\left(\theta \right)}\right)}{\omega \left(\theta \right)}\right)\Delta \theta \right)$$

(47)

Using (46) and (47) in (39), we get (45). This completes the proof. □

3. Conclusions

In this paper, with the help of the dynamic Jensen inequality, dynamic Hölder inequality and a simple consequence of Keller’s chain rule on time scales, we generalized a number of Opial-type inequalities to a general time scale. Besides that, in order to illustrate the theorems for each type of inequality applied to various time scales such as $\mathbb{R}$, $h\mathbb{Z}$, $\overline{q^{\mathbb{Z}}}$ and $\mathbb{Z}$ as a sub case of $h\mathbb{Z}$. For future studies, researchers may obtain some different generalizations for dynamic Opial inequality and its companion inequalities by using the results presented in this paper.