New Grüss type inequalities for double integrals

Abstract

In this paper, new Grüss type inequalities for double integrals are proved. Some sharp bounds are provided as well.

Introduction

For $a\text{,}b\text{,}c\text{,}d\in \mathbb{R}$, we consider the subset $\mathbb{D}≔\{(x\text{,}y):a⩽x⩽b\text{,}c⩽y⩽d\}\subseteq {\mathbb{R}}^2$. The Čebyšev functional

$$\mathcal{T}(f\text{,}g)≔\frac{1}{(b-a)(d-c)}{\int }_a^b{\int }_c^{d}f(t\text{,}s)g(t\text{,}s)\mathit{dsdt}-\frac{1}{(b-a)(d-c)}{\int }_a^b{\int }_c^{d}f(t\text{,}s)\mathit{dsdt}\frac{1}{(b-a)(d-c)}{\int }_a^b{\int }_c^{d}g(t\text{,}s)\mathit{dtds}$$

has interesting applications in the approximation of the integral of a product as pointed out in the references below.

In 2001, Hanna et al. [14] have proved the following inequality which is of Grüss type for double integrals, where f and g satisfy that

$$|f(x\text{,}y)-f(u\text{,}v)|⩽M_1|x-u{|}^{{\alpha }_1}+M_2|y-v{|}^{{\alpha }_2}\text{,}$$

where, $M_1\text{,}{M}_2>0\text{,}{\alpha }_1\text{,}{\alpha }_2\in (0\text{,}1]$,

$$|g(x\text{,}y)-g(u\text{,}v)|⩽N_1|x-u{|}^{{\beta }_1}+N_2|y-v{|}^{{\beta }_2}\text{,}$$

where, $N_1\text{,}{N}_2>0\text{,}{\beta }_1\text{,}{\beta }_2\in (0\text{,}1]$. Then,

$$|\mathcal{T}(f\text{,}g)|⩽M_1{N}_1\frac{{(b-a)}^{{\alpha }_1+{\beta }_1}}{({\alpha }_1+{\beta }_1+1)({\alpha }_1+{\beta }_1+2)}+M_1{N}_2\frac{2{(b-a)}^{{\alpha }_1}{(d-c)}^{{\beta }_2}}{({\alpha }_1+1)({\alpha }_1+2)({\beta }_2+1)({\beta }_2+2)}+M_2{N}_1\frac{2{(b-a)}^{{\alpha }_2}{(d-c)}^{{\beta }_1}}{({\alpha }_2+1)({\alpha }_2+2)({\beta }_1+1)({\beta }_1+2)}+M_2{N}_2\frac{{(b-a)}^{{\alpha }_2+{\beta }_2}}{({\alpha }_2+{\beta }_2+1)({\alpha }_2+{\beta }_2+2)}\text{.}$$

When ${\alpha }_1={\alpha }_2=1$ and ${\beta }_1={\beta }_2=1$, then

$$|f(x\text{,}y)-f(u\text{,}v)|⩽L_1|x-u|+L_2|y-v|\text{,}$$

$$|f(x\text{,}y)-f(u\text{,}v)|⩽K_1|x-u|+K_2|y-v|\text{,}$$

where, $L_1\text{,}{L}_2\text{,}{K}_1\text{,}{K}_2>0$ then (1.2) becomes

$$|\mathcal{T}(f\text{,}g)|⩽L_1{K}_1\frac{{(b-a)}^2}{12}+L_1{K}_2\frac{(b-a)(d-c)}{18}+L_2{K}_1\frac{(b-a)(d-c)}{18}+L_2{K}_2\frac{{(d-c)}^2}{12}\text{.}$$

After that, in 2002, Pachpatte [15] has established two inequalities of Grüss type involving continuous functions of two independent variables whose first and second partial derivatives are exist, continuous and belong to $L_{\infty }(\mathbb{D})$; for details see [15]. For more recent results about multivariate and multidimensional Grüss type inequalities the reader may refer to [1], [2], [3], [16], for another Grüss type inequalities see [5], [6] and [8], [9], [10], [11], [12].

Functions of bounded variation are of great interest and usefulness because of their valuable properties and multiple applications in several subfields including rectifiable curves, Fourier series, Stieltjes integrals, the calculus of variations and others. According to Clarkson and Adams [7], several definitions have been given under which a function of two or more independent variables shall be said to be of bounded variation. Among of these definitions, six are usually associated with the names of Vitali, Hardy, Arzelà, Pierpont, Fréchet, and Tonelli. For more details about these definitions, the interested reader may refer to [4], [7] and the recent book [13].

In this paper, we are mainly interested with the Arzelà definition, as follows:

Let $P:=\{(x_i\text{,}{y}_i):x_{i-1}⩽x⩽x_i\text{;}{y}_{i-1}⩽y⩽y_i\text{;}i=1\text{,}\dots \text{,}n\}$ be a partition of $\mathbb{D}$, write

$$\mathrm{\Delta }f(x_i\text{,}{y}_i)=f(x_i\text{,}{y}_i)-f(x_{i-1}\text{,}{y}_{i-1})$$

for $i=1\text{,}2\text{,}\dots \text{,}n$. The function $f(x\text{,}y)$ is said to be of bounded variation in the Arzelà sense (or simply bounded variation) if there exists a positive quantity M such that for every partition on $\mathbb{D}$ we have ${\sum }_{i=1}^n|\mathrm{\Delta }f(x_i\text{,}{y}_i)|⩽M$.

Therefore, one can define the concept of total variation of a function of two variables, as follows:

Let f be of bounded variation on $\mathbb{D}$, and let $\sum (P)$ denote the sum ${\sum }_{i=1}^n|\mathrm{\Delta }f(x_i\text{,}{y}_i)|$ corresponding to the partition P of $\mathbb{D}$. The number

$${\displaystyle \underset{\mathbb{D}}{\bigvee }}(f)≔\sup \left\{\sum P:P\in \mathcal{P}(\mathbb{D})\right\}$$

is called the total variation of f on $\mathbb{D}$.

The aim of this paper is to establish several new bounds for the Čebyšev function (1.1) under various assumptions.