Some Ostrowski and Cebysev type inequality in three variables on Time Scales

TL;DR

This paper investigates Ostrowski and Cebysev type inequalities involving three variables within the framework of Time Scales, unifying continuous and discrete analysis.

Contribution

It introduces new Ostrowski and Cebysev inequalities for three variables on Time Scales, extending existing inequalities to a more general setting.

Findings

Derived new inequalities on Time Scales for three variables

Unified continuous and discrete cases of inequalities

Provided potential applications in dynamic equations

Abstract

The main objective of this paper is to study some Ostrowski and Cebysev type inequalities in three variables on Time Scales

Equations

$$\left|f\left(x\right)-\frac{1}{b-a}\underset{a}{\overset{b}{\int }}f\left(t\right)dt\right|\le \left[\frac{1}{4}+\frac{{\left(x-\frac{a+b}{2}\right)}^2}{{\left(b-a\right)}^2}\right]\left(b-a\right){\Vert f^{\prime }\Vert }_{\infty },$$

$$\begin{array}[]{l}\left|{\frac{1}{{b-a}}\int\limits_{a}^{b}{f\left(x\right)g\left(x\right)dx}-\left({\frac{1}{{b-a}}\int\limits_{a}^{b}{f\left(x\right)dx}}\right)\left({\frac{1}{{b-a}}\int\limits_{a}^{b}{g\left(x\right)dx}}\right)}\right|\\ \leq\frac{1}{{12}}\left({b-a}\right)^{2}\left\|{f^{\prime}}\right\|_{\infty}\left\|{g^{\prime}}\right\|_{\infty},\\ \end{array}$$

$$\begin{array}[]{l}\left|{f\left({\sigma_{1}(t_{1}),t_{2},t_{3}}\right)-f(s,t_{2},t_{3})-f^{\Delta_{1}}\left({t_{1},t_{2},t_{3}}\right)\left({\sigma_{1}(t_{1})-s-\tau}\right)}\right|\\ \leq\varepsilon\left|{\sigma_{1}(t_{1})-s-\tau}\right|,\\ \end{array}$$

$$|\underset{a_1}{\overset{b_1}{\int }}\underset{a_2}{\overset{b_2}{\int }}\underset{a_2}{\overset{b_3}{\int }}f\left(x,y,z\right){\Delta }_{3}z{\Delta }_{2}y{\Delta }_{1}x$$

$$-\frac{1}{8}[\left(b_1-a_1\right)\left(b_2-a_2\right)\left(b_3-a_3\right)[f\left({\sigma }_1\left(s\right),{\sigma }_2\left(t\right),{\sigma }_3\left(\tau \right)\right)$$

$$+f\left(b_1,b_2,b_3\right)+f\left({\sigma }_1\left(s\right),{\sigma }_2\left(t\right),b_3\right)+f\left({\sigma }_1\left(s\right),b_2,{\sigma }_3\left(\tau \right)\right)$$

$$+f\left(b_1,{\sigma }_2\left(t\right),{\sigma }_3\left(\tau \right)\right)+f\left({\sigma }_1\left(s\right),b_2,b_3\right)+f\left(b_1,b_2,{\sigma }_3\left(\tau \right)\right)$$

$$+f\left(b_1,{\sigma }_2\left(t\right),b_3\right)]]$$

$$+\frac{1}{4}\left(b_2-a_2\right)\left(b_3-a_3\right)\underset{a_1}{\overset{b_1}{\int }}[f\left(x,{\sigma }_2\left(t\right),{\sigma }_3\left(\tau \right)\right)+f\left(x,c,d\right)$$

$$+f\left(x,b_2,{\sigma }_3\left(\tau \right)\right)+f\left(x,{\sigma }_2\left(t\right),b_3\right)]{\Delta }_{1}x$$

$$+\frac{1}{4}\left(b_1-a_1\right)\left(b_2-a_2\right)\underset{a_3}{\overset{b_3}{\int }}[f\left({\sigma }_1\left(s\right),{\sigma }_2\left(t\right),z\right)+f\left(b_1,b_2,z\right)$$

$$+f\left(b_1,{\sigma }_2\left(t\right),z\right)+f\left({\sigma }_1\left(s\right),c,z\right)]{\Delta }_{3}z$$

$$-\frac{1}{2}\left(b_1-a_1\right)\underset{a_2}{\overset{b_2}{\int }}\underset{a_3}{\overset{b_3}{\int }}\left[f\left({\sigma }_1\left(s\right),y,z\right)+f\left(b_1,y,z\right)\right]{\Delta }_{3}z{\Delta }_{2}y$$

$$-\frac{1}{2}\left(b_2-a_2\right)\underset{a_2}{\overset{b_2}{\int }}\underset{a_3}{\overset{b_3}{\int }}\left[f\left(x,{\sigma }_2\left(t\right),z\right)+f\left(x,b_2,z\right)\right]{\Delta }_{3}z{\Delta }_{2}y$$

$$-\frac{1}{2}\left(b_3-a_3\right)\underset{a_1}{\overset{b_1}{\int }}\underset{a_2}{\overset{b_2}{\int }}\left[f\left(x,y,{\sigma }_3\left(\tau \right)\right)+f\left(x,y,b_3\right)\right]{\Delta }_{2}y{\Delta }_{1}x$$

$$\le \underset{a_1}{\overset{b_1}{\int }}\underset{a_2}{\overset{b_2}{\int }}\underset{a_3}{\overset{b_3}{\int }}\left|\frac{{\partial }^{3}f\left(\alpha ,\beta ,\gamma \right)}{{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha }\right|{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha .$$

$$\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{y}{\int }}\underset{{\sigma }_3\left(\tau \right)}{\overset{z}{\int }}\frac{{\partial }^{3}f\left(\alpha ,\beta ,\gamma \right)}{{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha }{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha$$

$$={\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{y}{\int }}\frac{{\partial }^{2}f\left(\alpha ,\beta ,z\right)}{{\Delta }_2\beta {\Delta }_1\alpha }|}_{{\sigma }_3\left(\tau \right)}^z{\Delta }_2\beta {\Delta }_1\alpha$$

$$=\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{y}{\int }}\frac{{\partial }^{2}f\left(\alpha ,\beta ,z\right)}{{\Delta }_2\beta {\Delta }_1\alpha }{\Delta }_2\beta {\Delta }_1\alpha -\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{y}{\int }}\frac{{\partial }^{2}f\left(\alpha ,\beta ,{\sigma }_3\left(\tau \right)\right)}{{\Delta }_2\beta {\Delta }_1\alpha }{\Delta }_2\beta {\Delta }_1\alpha$$

$$=\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}{\frac{\partial f\left(\alpha ,\beta ,z\right)}{{\Delta }_1\alpha }|}_{{\sigma }_2\left(t\right)}^y{\Delta }_1\alpha -\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}{\frac{\partial f\left(\alpha ,\beta ,{\sigma }_3\left(\tau \right)\right)}{{\Delta }_1\alpha }|}_{{\sigma }_2\left(t\right)}^y{\Delta }_1\alpha$$

$$=\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\frac{\partial f\left(\alpha ,y,z\right)}{{\Delta }_1\alpha }{\Delta }_1\alpha -\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\frac{\partial f\left(\alpha ,{\sigma }_2\left(t\right),z\right)}{{\Delta }_1\alpha }{\Delta }_1\alpha$$

$$-\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\frac{\partial f\left(\alpha ,y,{\sigma }_3\left(\tau \right)\right)}{{\Delta }_1\alpha }{\Delta }_1\alpha +\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\frac{\partial f\left(\alpha ,{\sigma }_2\left(t\right),{\sigma }_3\left(\tau \right)\right)}{{\Delta }_1\alpha }{\Delta }_1\alpha$$

$$={f\left(\alpha ,y,z\right)|}_{{\sigma }_1\left(s\right)}^x-{f\left(\alpha ,{\sigma }_2\left(t\right),z\right)|}_{{\sigma }_1\left(s\right)}^x-{f\left(\alpha ,y,{\sigma }_3\left(\tau \right)\right)|}_{{\sigma }_1\left(s\right)}^x$$

$$+{f\left(\alpha ,{\sigma }_2\left(t\right),{\sigma }_3\left(t\right)\right)|}_{{\sigma }_1\left(s\right)}^x$$

$$=f(x,y,z)-f({\sigma }_1\left(s\right),y,z)-f\left(x,{\sigma }_2\left(t\right),z\right)$$

$$+f({\sigma }_1\left(s\right),{\sigma }_2\left(t\right),z)-f(x,y,{\sigma }_3\left(\tau \right))+f\left({\sigma }_1\left(s\right),y,{\sigma }_3\left(\tau \right)\right)$$

$$+f\left(x,{\sigma }_2\left(t\right),{\sigma }_3\left(\tau \right)\right)-f\left({\sigma }_1\left(s\right),{\sigma }_2\left(t\right),{\sigma }_3\left(\tau \right)\right),$$

$$f\left(x,y,z\right)=f({\sigma }_1\left(s\right),y,z)+f\left(x,{\sigma }_2\left(t\right),z\right)+f(x,y,{\sigma }_3\left(\tau \right))$$

$$+f\left({\sigma }_1\left(s\right),{\sigma }_2\left(t\right),{\sigma }_3\left(\tau \right)\right)-f({\sigma }_1\left(s\right),{\sigma }_2\left(t\right),z)-f\left({\sigma }_1\left(s\right),y,{\sigma }_3\left(\tau \right)\right)$$

$$-f\left(x,{\sigma }_2\left(t\right),{\sigma }_3\left(\tau \right)\right)+\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{y}{\int }}\underset{{\sigma }_3\left(\tau \right)}{\overset{z}{\int }}\frac{{\partial }^{3}f\left(\alpha ,\beta ,\gamma \right)}{{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha }{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha .$$

$$f\left(x,y,z\right)=f(x,y,b_3)+f\left(x,{\sigma }_2\left(t\right),z\right)+f({\sigma }_1\left(s\right),{\sigma }_2\left(t\right),b_3)$$

$$-f({\sigma }_1\left(s\right),y,b_3)+f({\sigma }_1\left(s\right),y,z)-f(x,{\sigma }_2\left(t\right),b_3)$$

$$-f\left({\sigma }_1\left(s\right),{\sigma }_2\left(t\right),z\right)-\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{y}{\int }}\underset{z}{\overset{b_3}{\int }}\frac{{\partial }^{3}f\left(\alpha ,\beta ,\gamma \right)}{{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha }{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha ,$$

$$f\left(x,y,z\right)=f\left(x,b_2,z\right)+f\left({\sigma }_1\left(s\right),b_2,{\sigma }_3\left(\tau \right)\right)+f\left(x,y,{\sigma }_3\left(\tau \right)\right)$$

$$+f\left({\sigma }_1\left(s\right),y,z\right)-f\left({\sigma }_1\left(s\right),b_2,z\right)-f\left(x,b_2,{\sigma }_3\left(\tau \right)\right)$$

$$-f\left({\sigma }_1\left(s\right),y,{\sigma }_3\left(\tau \right)\right)-\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\underset{y}{\overset{b_2}{\int }}\underset{{\sigma }_3\left(\tau \right)}{\overset{z}{\int }}\frac{{\partial }^{3}f\left(\alpha ,\beta ,\gamma \right)}{{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha }{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha ,$$

$$f\left(x,y,z\right)=f\left(b_1,y,z\right)+f\left(x,{\sigma }_2\left(t\right),z\right)+f\left(x,y,{\sigma }_3\left(\tau \right)\right)$$

$$+f\left(b_1,{\sigma }_2\left(t\right),{\sigma }_3\left(\tau \right)\right)-f\left(b_1,{\sigma }_2\left(t\right),z\right)-f\left(b,y,{\sigma }_3\left(\tau \right)\right)$$

$$-f\left(x,{\sigma }_2\left(t\right),{\sigma }_3\left(\tau \right)\right)-\underset{x}{\overset{b_1}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{y}{\int }}\underset{{\sigma }_3\left(\tau \right)}{\overset{z}{\int }}\frac{{\partial }^{3}f\left(\alpha ,\beta ,\gamma \right)}{{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha }{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha ,$$

$$f\left(x,y,z\right)=f\left({\sigma }_1\left(s\right),b_2,b_3\right)+f\left(x,y,b_3\right)+f\left(x,b_2,z\right)$$

$$+f\left({\sigma }_1\left(s\right),y,z\right)-f\left(x,b_2,b_3\right)-f\left({\sigma }_1\left(s\right),y,b_3\right)$$

$$-f\left({\sigma }_1\left(s\right),b_2,z\right)+\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\underset{y}{\overset{b_2}{\int }}\underset{z}{\overset{b_3}{\int }}\frac{{\partial }^{3}f\left(\alpha ,\beta ,\gamma \right)}{{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha }{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha ,$$

$$f\left(x,y,z\right)=f\left(x,b_2,z\right)+f\left(b_1,y,z\right)+f\left(b_1,b_2,{\sigma }_3\left(\tau \right)\right)$$

$$+f\left(x,y,{\sigma }_3\left(\tau \right)\right)-f\left(b_1,b_2,z\right)-f\left(x,b_2,{\sigma }_3\left(\tau \right)\right)$$

$$-f\left(b_1,y,{\sigma }_3\left(\tau \right)\right)+\underset{x}{\overset{b_1}{\int }}\underset{y}{\overset{b_2}{\int }}\underset{{\sigma }_3\left(\tau \right)}{\overset{z}{\int }}\frac{{\partial }^{3}f\left(\alpha ,\beta ,\gamma \right)}{{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha }{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha ,$$

$$f\left(x,y,z\right)=f\left(x,y,b_3\right)+f\left(b_1,{\sigma }_2\left(t\right),b_3\right)+f\left(b_1,y,z\right)$$

$$+f\left(x,{\sigma }_2\left(t\right),z\right)-f\left(b_1,y,b_3\right)-f\left(x,{\sigma }_2\left(t\right),b_3\right)$$

$$-f\left(b_1,{\sigma }_2\left(t\right),y\right)+\underset{x}{\overset{b_1}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{y}{\int }}\underset{z}{\overset{b_3}{\int }}\frac{{\partial }^{3}f\left(\alpha ,\beta ,\gamma \right)}{{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha }{\Delta }_3\gamma {\Delta }_2\beta {\Delta }_1\alpha ,$$

Full text

1. Introduction

In 1938 A.M. Ostrowski [21] proved the inequality: Let $f:[a,b]\to\mathbb{R}$ be continuous on [a,b] and differentiable on $(a,b)$ with derivative $f^{\prime}:[a,b]\to\mathbb{R}$ being bounded on $(a,b)$, that is $\left\|{f^{\prime}}\right\|_{\infty}=\sup_{x\in\left({a,b}\right)}\left|{f^{\prime}\left(x\right)}\right|<\infty$. Then

[TABLE]

for all $x\in[a,b]$. In this the inequality gives an upper bound for the approximation of the integral average $\frac{1}{{b-a}}\int\limits_{a}^{b}{f\left(t\right)dt}$ by the value $f(x)$ at the point $x\in[a,b]$.

In 1882, P.L. Cebysev [9] proved the following inequality:

[TABLE]

provided $f,g$ are absolutely continuous functions defined on [a,b] and $f^{\prime},g^{\prime}\in L_{\infty}\left[{a,b}\right]$.

Recently many researchers have obtained various generalizations, extensions and variants of the above inequalities[2,4-5,8, 10-20,22-24] on time scale calculus. German mathematician Stefan Hilger has initiated the study of time scales calculus which unifies the theory of both differential and difference calculus [10]. The basic information on time scales can be found in [1, 3, 6, 7, 10] Motivated by the above literature in this paper we present some dynamic Ostrowski and Cebysev type inequalities on time scales in three variables

In what follows $\mathbb{R}$ denotes the set of real number and $\mathbb{T}$ is nonempty closed subset of $\mathbb{R}$. For $t\in\mathbb{T}$ the mapping $\sigma,\rho:\mathbb{T}\to\mathbb{T}$ are defined by $\sigma\left(t\right)=\inf\left\{{s\in\mathbb{T}:s>t}\right\}$ and $\rho\left(t\right)=\sup\left\{{s\in\mathbb{T}:s<t}\right\}$ are called the forward and backward jump operators respectively. We define $f:\mathbb{T}\to\mathbb{R}$ is rd-continuous provided f is continuous at each right dense point of $\mathbb{T}$ and has left sided limit at each dense point on $\mathbb{T}$. Now we denote $C_{rd}$ for the set of rd-continuous function defined on $\mathbb{T}$. Let $\mathbb{T}_{1},\mathbb{T}_{2},\mathbb{T}_{3}$ be three time scales. Let $\sigma_{i},\rho_{i}$ and $\Delta_{i}$ for $i=1,2,3$ denotes the forward jump operator, backward jump operator and delta differentiation operator respectively on $\mathbb{T}_{1},\mathbb{T}_{2}$ and $\mathbb{T}_{3}$. Let $a_{i}<b_{i}$ be points in $\mathbb{T}_{i}$ and $[a_{i},b_{i})$ are half closed bounded intervals in $\mathbb{T}_{i}$.

We say that a real valued function $f$ on $\mathbb{T}_{1}\times\mathbb{T}_{2}\times\mathbb{T}_{3}$ has a $\Delta_{1}$ partial derivative $f^{\Delta_{1}}\left({t_{1},t_{2},t_{3}}\right)$ with respect to $t_{1}$ if for each $\epsilon>0$ there exists a neighborhood $U_{t_{1}}$ of $t_{1}$ such that

[TABLE]

for each $s\in U_{t_{1}}$, $t_{2}\in\mathbb{T}_{2}$ and $t_{3}\in\mathbb{T}_{3}$.

If $f(x,y,z)$ is delta differential function defined on $\left[{a_{1},b_{1}}\right]\times\left[{a_{2},b_{2}}\right]\times\left[{a_{3},b_{3}}\right]$ for $a_{1},b_{1},a_{2},b_{2},a_{3},b_{3}$ in $\mathbb{R}^{+}$. Then its partial delta derivative are defined by

$f^{\Delta_{1}}\left({x,y,z}\right)=\frac{\partial}{{\partial x}}f\left({x,y,z}\right),f^{\Delta_{2}}\left({x,y,z}\right)=\frac{\partial}{{\partial y}}f\left({x,y,z}\right),f^{\Delta_{3}}\left({x,y,z}\right)=\frac{\partial}{{\partial z}}f\left({x,y,z}\right)$.

Let $CC_{rd}$ denotes the set of all rd-continuous functions and let $CC^{\prime}_{rd}$ denotes the set of all functions for which $\Delta_{1}$ partial derivative, $\Delta_{2}$ partial derivative and $\Delta_{3}$ partial derivative exists and are in $CC_{rd}$.

2. Ostrowski Inequality In Three Variables on Time Scales

Now we give dynamic Ostrowski Inequality in three Variables on Time Scales

Theorem 2.1

Let $f\in CC_{rd}\left({\left[{a_{1},b_{1}}\right]\times\left[{a_{2},b_{2}}\right]\times\left[{a_{3},b_{3}}\right],\mathbb{R}}\right)$

[TABLE]

Proof.

: From the hypotheses we have

[TABLE]

that is

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Similarly we have

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[TABLE]

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Integrating the above equations(2.2-2.9) and adding them we get the required inequality (2.1).

If we have $\mathbb{T}=\mathbb{R}$ in above theorem we get the continuous version to the above inequality.

Corollary 2.2

Let $f:\left[{a_{1},b_{1}}\right]\times\left[{a_{2},b_{2}}\right]\times\left[{a_{3},b_{3}}\right]\to\mathbb{R}$ be differentiable and continuous function. Then

[TABLE]

If we have $\mathbb{T}=\mathbb{Z}$ in above theorem we get the discrete version of above inequality

Corollary 2.3

Let $\mathbb{Z}_{1}=\left\{{1,2,...,b_{1}+1}\right\}$,$\mathbb{Z}_{2}=\left\{{1,2,...,b_{2}+1}\right\}$,$\mathbb{Z}_{3}=\left\{{1,2,...,b_{3}+1}\right\}$ for $b_{1},b_{2},b_{3}\in\mathbb{Z}$ and $f:\mathbb{Z}_{1}\times\mathbb{Z}_{2}\times\mathbb{Z}_{3}\to\mathbb{R}$

[TABLE]

3. Cebysev type inequalities in three Variables

Now we give the Cebysev type inequalities in three Variables on time scales.

We use following notations to simplify the details.

[TABLE]

and

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Now we give Cebysev type inequality on time scales involving functions in three variables.

Theorem 3.1

Let $f,g\in CC_{rd}\left({\left[{a_{1},b_{1}}\right]\times\left[{a_{2},b_{2}}\right]\times\left[{a_{3},b_{3}}\right],\mathbb{R}}\right)$ be rd-continuous function and $f^{\Delta_{3}\Delta_{2}\Delta_{1}}\left({x,y,z}\right)$ and $g^{\Delta_{3}\Delta_{2}\Delta_{1}}\left({x,y,z}\right)$ exist and rd-continuous and bounded. Then

[TABLE]

Proof.

: Adding $(2.2)-(2.9)$ and using the notations in equation $(3.1)$ and $(3.2)$ we have

[TABLE]

for $(x,y,z)\in[a_{1},b_{1}]\times[a_{2},b_{2}]\times[a_{3},b_{3}]$.

Similarly for $g$ we have

[TABLE]

Multiplying $(3.4)$ by $g(x,y,z)$ and $(3.5)$ by $f(x,y,z)$ and adding resultant identities and then integrating we have

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From the properties of modulus and integrals we have

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Similarly we have

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Now from $(3.6),(3.7),(3.8)$ we have

[TABLE]

which is required inequality $(3.2)$. If we have $\mathbb{T}=\mathbb{R}$ in above theorem we have

Corollary 3.2

Let $f,g:\left[{a_{1},b_{1}}\right]\times\left[{a_{2},b_{2}}\right]\times\left[{a_{3},b_{3}}\right]\to\mathbb{R}$ be differentiable and continuous function and $D_{3}D_{2}D_{1}f\left({x,y,z}\right)$, $D_{3}D_{2}D_{1}g\left({x,y,z}\right)$ exist continuous and bounded. Then

[TABLE]

where

[TABLE]

If we have $\mathbb{T}=\mathbb{Z}$ in above theorem we have discrete version of above inequality.

Corollary 3.3

Let $Z_{1}=\left\{{1,2,...,k+1}\right\}$,$\mathbb{Z}_{2}=\left\{{1,2,...,m+1}\right\}$,$\mathbb{Z}_{3}=\left\{{1,2,...,n+1}\right\}$ for $b_{1},b_{2},b_{3}\in\mathbb{Z}$ and $f,g:\mathbb{Z}_{1}\times\mathbb{Z}_{2}\times\mathbb{Z}_{3}\to\mathbb{R}$ be functions such that $\Delta_{3}\Delta_{2}\Delta_{1}f\left({x,y,z}\right)$, $\Delta_{3}\Delta_{2}\Delta_{1}g\left({x,y,z}\right)$ exist and are bounded. Then

[TABLE]

where

[TABLE]

Acknowledgment

This research is supported by Science and Engineering Research Board (SERB, New Delhi, India) Project File No. SB/S4/MS:861/13.