Some Ostrowski type inequalities for double integrals On Time Scales

TL;DR

This paper investigates Ostrowski and Trapezoid type inequalities for double integrals within the framework of Time Scales, extending classical inequalities to a more general setting.

Contribution

It introduces new Ostrowski and Trapezoid inequalities for double integrals on Time Scales, broadening the scope of integral inequalities in this unified calculus.

Findings

Derived new inequalities for double integrals on Time Scales.

Extended classical inequalities to a more general setting.

Presented additional related inequalities.

Abstract

The main objective of this paper is to study some Ostrowski and Trapezoid type inequalities for double integrals on Time Scales. Some other interesting inequalities are also given.

Equations

$$\left|f\left({\sigma }_1\left(t_1\right),t_2\right)-f\left(s,t_2\right)-f^{{\Delta }_1}\left(t_1,t_2\right)\left({\sigma }_1\left(t_1\right)-s\right)\right|\le \epsilon \left|{\sigma }_1\left(t_1\right)-s\right|,$$

$$\left|f\left(t_1,{\sigma }_2\left(t_2\right)\right)-f\left(t_1,l\right)-f^{{\Delta }_2}\left(t_1,t_2\right)\left({\sigma }_2\left(t_2\right)-l\right)\right|\le \eta \left|{\sigma }_2\left(t_2\right)-l\right|,$$

$$|\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}\{[f\left(x,y\right)g\left(x,y\right)-\frac{1}{2}[P\left(f\left(x,y\right)\right)g\left(x,y\right)$$

$$+P\left(g\left(x,y\right)\right)f\left(x,y\right)]\}{\Delta }_{2}y{\Delta }_{1}x|$$

$$\le \frac{1}{8}\left(b-a\right)\left(d-c\right)\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}[|g\left(x,y\right)|{\Vert f^{{\Delta }_2{\Delta }_1}\Vert }_{\infty }$$

$${+f\left(x,y\right)\Vert g^{{\Delta }_2{\Delta }_1}\Vert ]}_{\infty }{\Delta }_{2}y{\Delta }_{1}x.$$

$$P\left(f\left(x,y\right)\right)$$

$$-\frac{1}{4}\left[f\left({\sigma }_1\left(s\right),{\sigma }_2\left(t\right)\right)+f\left({\sigma }_1\left(s\right),d\right)+f\left(b,{\sigma }_2\left(t\right)\right)+f\left(b,d\right)\right],$$

$$Q\left(f^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right)$$

$$=\underset{{\sigma }_1\left(s\right)}{\overset{b}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta -\underset{{\sigma }_1\left(s\right)}{\overset{b}{\int }}\underset{y}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta$$

$$-\underset{x}{\overset{b}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta +\underset{x}{\overset{b}{\int }}\underset{y}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta .$$

$$\underset{{\sigma }_1\left(s\right)}{\overset{b}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta$$

$$=\underset{{\sigma }_1\left(s\right)}{\overset{x}{\int }}\left[{\frac{\partial f\left(\eta ,\tau \right)}{{\Delta }_1\eta }|}_{{\sigma }_2\left(t\right)}^y\right]{\Delta }_1\eta$$

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

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

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

$$\underset{{\sigma }_1\left(s\right)}{\overset{b}{\int }}\underset{y}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta$$

$$=-f\left(x,y\right)-f\left({\sigma }_1\left(s\right),d\right)+f\left(x,d\right)+f\left({\sigma }_1\left(s\right),y\right),$$

$$\underset{x}{\overset{b}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta$$

$$=-f\left(x,y\right)-f\left(b,{\sigma }_2\left(t\right)\right)+f\left(x,{\sigma }_2\left(t\right)\right)+f\left(b,{\sigma }_2\left(t\right)\right),$$

$$\underset{x}{\overset{b}{\int }}\underset{y}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta$$

$$4f\left(x,y\right)-2f\left({\sigma }_1\left(s\right),y\right)-2f\left(x,{\sigma }_2\left(t\right)\right)-2f\left(x,d\right)-2f\left(b,y\right)$$

$$+f\left({\sigma }_1\left(s\right),{\sigma }_2\left(t\right)\right)+f\left({\sigma }_1\left(s\right),d\right)+f\left(b,{\sigma }_2\left(t\right)\right)+f\left(b,d\right)$$

$$=\underset{{\sigma }_1\left(s\right)}{\overset{b}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta -\underset{{\sigma }_1\left(s\right)}{\overset{b}{\int }}\underset{y}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta$$

$$-\underset{x}{\overset{b}{\int }}\underset{{\sigma }_2\left(t\right)}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta +\underset{x}{\overset{b}{\int }}\underset{y}{\overset{d}{\int }}\frac{{\partial }^{2}f\left(\eta ,\tau \right)}{{\Delta }_2\tau {\Delta }_1\eta }{\Delta }_2\tau {\Delta }_1\eta .$$

$$f\left(x,y\right)-P\left(f\left(x,y\right)\right)=\frac{1}{4}Q\left(f^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right),$$

$$g\left(x,y\right)-P\left(g\left(x,y\right)\right)=\frac{1}{4}Q\left(g^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right),$$

$$2f\left(x,y\right)g\left(x,y\right)-g\left(x,y\right)P\left(f\left(x,y\right)\right)-f(x,y)P\left(g\left(x,y\right)\right)$$

$$=\frac{1}{4}g\left(x,y\right)Q\left(f^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right)+\frac{1}{4}f\left(x,y\right)Q\left(g^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right).$$

$$\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}[f\left(x,y\right)g\left(x,y\right)$$

$$-\frac{1}{2}\left[P\left(f\left(x,y\right)\right)g\left(x,y\right)+P\left(g\left(x,y\right)\right)f\left(x,y\right)\right]]{\Delta }_{2}y{\Delta }_{1}x$$

$$=\frac{1}{8}\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}\left[Q\left(f^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right)g\left(x,y\right)+Q\left(g^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right)f\left(x,y\right)\right]{\Delta }_{2}y{\Delta }_{1}x.$$

$$\left|Q\left(f^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right)\right|$$

$$\le {\Vert f^{{\Delta }_2{\Delta }_1}\Vert }_{\infty }\left(b-a\right)\left(d-c\right),$$

$$\left|Q\left(g^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right)\right|$$

$$\le {\Vert g^{{\Delta }_2{\Delta }_1}\Vert }_{\infty }\left(b-a\right)\left(d-c\right).$$

$$|\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}[f\left(x,y\right)g\left(x,y\right)$$

$$-\frac{1}{2}\left[P\left(f\left(x,y\right)\right)g\left(x,y\right)+P\left(g\left(x,y\right)\right)f\left(x,y\right)\right]]{\Delta }_{2}y{\Delta }_{1}x|$$

$$\le \frac{1}{8}\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}[\left|g\left(x,y\right)\right|\left|Q\left(f^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right)\right|$$

$$+\left|f\left(x,y\right)\right|\left|Q\left(g^{{\Delta }_2{\Delta }_1}\left(x,y\right)\right)\right|]{\Delta }_{2}y{\Delta }_{1}x$$

$$\le \frac{1}{8}\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}[\left|g\left(x,y\right)\right|\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}\left|f^{{\Delta }_2{\Delta }_1}\left(t,s\right)\right|{\Delta }_{2}s{\Delta }_{1}t$$

$$+\left|f\left(x,y\right)\right|\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}\left|g^{{\Delta }_2{\Delta }_1}\left(t,s\right)\right|{\Delta }_{2}s{\Delta }_{1}t]{\Delta }_{2}y{\Delta }_{1}x$$

$$\le \frac{1}{8}\left(b-a\right)\left(d-c\right)\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}[\left|g\left(x,y\right)\right|{\Vert f^{{\Delta }_2{\Delta }_1}\Vert }_{\infty }$$

Full text

1. Introduction

In year 1988 the German mathematicain in his Ph.D dissertation has initiated the study of time scales calculus which unifies the theory of both differential and difference calculus [9]. Dynamical equations and inequality’s can be used studying various properties and model many phenomena in economics [5], biological systems [23] and various systems in neural network [11].

In [8] Bohner and Matthews have given the Ostrowski inequality and Montgomery identity on time scales. Some results on Ostrowski and Gruss inequality were obtained by N. Ahmad, W. Liu and others [2, 12, 22]. Recently in [4, 10, 13, 15, 24] authors have obtained some new Ostrowski type inequalities. Weighted Ostrowski and Trapezoid inequalities on time scales are obtained by W. Liu and others in [16, 17, 18]. In [21] M. Sarikya have studied some weighted Ostrowski and Chebsev type inequalities on time scales. Motivated by the results in the above paper we obtain some Ostrowski and Trapezoid type inequalities for double integrals on time scales.

In what follows the time scale $\mathbb{T}$ is a nonempty closed subset of $\mathbb{R}$. Let $t\in\mathbb{T}$ the mapping $\sigma,\rho:\mathbb{T}\to\mathbb{T}$ are defined as $\sigma\left(t\right)=\inf\left\{{s\in\mathbb{T}:s>t}\right\}$ and $\rho\left(t\right)=\sup\left\{{s\in\mathbb{T}:s<t}\right\}$ are called the forward and backward jump operators respectively.

We say that $f:\mathbb{T}\to\mathbb{R}$ is rd-continuous provided $f$ is continuous at each right-dense point of $\mathbb{T}$ and has a finite left sided limit at each left dense point of $\mathbb{T}$. $C_{rd}$ denotes the set of rd-continuous function defined on $\mathbb{T}$. Let $\mathbb{T}_{1}$ and $\mathbb{T}_{2}$ be two time scales with at least two points and consider the time scales intervals $\overline{\mathbb{T}}_{1}=\left[{x_{0},\infty}\right)\cap\mathbb{T}_{1}$ and $\overline{\mathbb{T}}_{2}=\left[{y_{0},\infty}\right)\cap\mathbb{T}_{2}$ for $x_{0}\in\mathbb{T}_{1}$ and $y_{0}\in\mathbb{T}_{2}$ and $\Omega=\mathbb{T}_{1}\times\mathbb{T}_{2}$. Let $\sigma_{1},\rho_{1},\Delta_{1}$ and $\sigma_{2},\rho_{2},\Delta_{2}$ denote the forward jump operators, backward jump operators and the delta differentiation operator respectively on $\mathbb{T}_{1}$ and $\mathbb{T}_{2}$. Let $a<b$ be points in $\mathbb{T}_{1}$, $c<d$ are point in $\mathbb{T}_{2}$, $[a,b)$ is the half closed bounded interval in $\mathbb{T}_{1}$, and $[c,d)$ is the half closed bounded interval in $\mathbb{T}_{2}$.

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

[TABLE]

for all $l\in U_{t_{2}}$, $t_{1}\in\mathbb{T}_{1}$. The function $f$ is called rd-continuous in $t_{2}$ if for every $\alpha_{1}\in\mathbb{T}_{1}$, the function $f(\alpha_{1},.)$ is rd-continuous on $\mathbb{T}_{2}$. The function $f$ is called rd-continuous in $t_{1}$ if for every $\alpha_{2}\in\mathbb{T}_{2}$ the function $f(.,\alpha_{2})$ is rd-continuous on $\mathbb{T}_{1}$.

Let $CC_{rd}$ denote the set of functions $f(t_{1},t_{2})$ on $\mathbb{T}_{1}\times\mathbb{T}_{2}$ where $f$ is rd continuous in $t_{1}$ and $t_{2}$. Let $CC^{\prime}_{rd}$ deontes the set of all functions $CC_{rd}$ for which both the $\Delta_{1}$ partial derivative and $\Delta_{2}$ partial derivative exists and are in $CC_{rd}$.

The basic information on time scales and inequalities can be found in [1, 3, 6, 7].

2. **Ostrowski Inequalities for double integrals on time scales **

Now we give Ostrowski Inequalities for double integrals on time scales

Theorem 2.1

Let $f,g\in CC^{\prime}_{rd}\left({\left[{a,b}\right]\times\left[{c,d}\right],\mathbb{R}}\right)$ and $f^{\Delta_{2}\Delta_{1}}\left({x,y}\right)$, $g^{\Delta_{2}\Delta_{1}}\left({x,y}\right)$ exist rd-continuous on $[a,b]\times[c,d]$. Then

[TABLE]

where

[TABLE]

and

[TABLE]

Similarly $P(g(x,y))$ and $Q\left({f^{\Delta_{2}\Delta_{1}}\left({x,y}\right)}\right)$ are defined similar to $(2.2)$ and $(2.3)$.

Proof.

From the hypotheses we have for $(x,y)\in[a,b]\times[c,d]$

[TABLE]

Similarly we have

[TABLE]

[TABLE]

and

[TABLE]

Adding above identities we have

[TABLE]

From $(2.3)$, $(2.4)$ and $(2.8)$ we have

[TABLE]

for $x,y\in[a,b]\times[c,d]$.

Similarly for function $g$ we have

[TABLE]

for $x,y\in[a,b]\times[c,d]$. Multiplying $(2.9)$ and $(2.10)$ by $g(x,y)$ and $f(x,y)$ and adding the resulting identities we get

[TABLE]

Integrating $(2.11)$ over $[a,b]\times[c,d]$ we have

[TABLE]

From the properties of modulus we have

[TABLE]

and

[TABLE]

From $(2.12),(2.13)$ and $(2.14)$ we have

[TABLE]

Which is required inequality.

Now we give the continuous and Discrete equivalent version of above inequality where $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{Z}$

Corollary 2.1

(Continuous Case) If we put $\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{R}$ we have

[TABLE]

[TABLE]

[TABLE]

Similarly $P(g(x,y))$ and $Q(D_{2}D_{1}g(x,y))$, which is Ostrowski inequality for Double integral.

Corollary 2.2

(Discrete Case) If $\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{Z}$ and $a=c=0$, $b=k\in\mathbb{N}$ and $d=r\in\mathbb{N}$. Then

[TABLE]

[TABLE]

[TABLE]

Which is Discrete Ostrowski Inequality.

Now we give some Ostrowski inequality for double integrals.

Theorem 2.2

Let $f,g,P(f(x,y)),P(g(x,y),f^{\Delta_{2}\Delta_{1}},g^{\Delta_{2}\Delta_{1}}$ be as in Theorem $2.1$ then

[TABLE]

for $[x,y]\in[a,b]\times[c,d]$.

Proof.

From $(2.9)$ and $(2.10)$ we have

[TABLE]

and

[TABLE]

for $[x,y]\in[a,b]\times[c,d]$.

Multiplying left hand side and right hand side of $(2.17)$ and $(2.18)$ we get

[TABLE]

Integrating $(2.19)$ over $[a,b]\times[c,d]$ and from the properties of modulus we have

[TABLE]

Now using $(2.13)$ and $(2.14)$ in $(2.15)$ we get the required inequality $(2.16)$.

Now we give continuous and discrete version of the inequality $(2.16)$ where $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{Z}$ which is as follows

Corollary 2.3

(Continuous Case) If we put $\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{R}$ in above we get

[TABLE]

where $f,g,P,Q,D_{2}D_{1}f,D_{2}D_{1}g$ is as in Corollary $2.1$. Which is Ostrowksi type inequality for double integral.

Corollary 2.4

(Discrete Case) If $\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{Z}$ and $a=c=0$,$b=k\in N$ and $d=r\in N$. Then

[TABLE]

where $P,Q$ are as in Corollary $2.2$. Which is discrete Ostrowski type inequality.

3. Trapezoid type Inequality on time scales

Now we give the dynamic Trapezoid type inequality on time scales.

Theorem 3.1

Let $f,f^{\Delta_{2}\Delta_{1}}$ be as in Theorem $2.1$. Then

[TABLE]

Proof.

From the proof of Theorem $2.1$ we have

[TABLE]

for $[x,y]\in[a,b]\times[c,d]$.

Integrating $(3.2)$ over $[a,b]\times[c,d]$ we get

[TABLE]

From the property of modulus and integrals we have

[TABLE]

From $(3.3)$ and $(3.4)$ we have

[TABLE]

which is required inequality. Now we give the Continuous and discrete version of Trapezoid inequality when $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{Z}$.

Corollary 3.1

(Continuous Case) If we put $\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{R}$ in above we get

[TABLE]

which is Continuous Trapezoid type inequality

Corollary 3.2

(Discrete Case) If $\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{Z}$ then we get

[TABLE]

which is Discrete Trapezoid type inequality

Acknowledgment

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