Abstract

In this paper, we first obtain the useful identity for double integrals via fractional integral operators. Then with the help of this identity we outline some significant Ostrowski type integral inequalities for functions in two variables. In accordance with this purpose we benefit from the properties of bounded function and concave mappings on co-ordinates. The established results are extensions of some existing Ostrowski type inequalities in the previous published studies.

1. Introduction

The study of various types of integral inequalities has been the focus of great attention for well over a century by a number of mathematicians, interested both in pure and applied mathematics. One of the many fundamental mathematical discoveries of A. M. Ostrowski ¹ is the following classical integral inequality associated with the differentiable mappings:

Theorem 1. Let be a differentiable mapping on whose derivative is bounded on i.e. Then, we have the inequality

for all

The constant is the best possible.

Ostrowski inequality has applications in quadrature, probability and optimization theory, stochastic, statistics, information and integral operator theory. During the past few years, a number of scientists have focused on Ostrowski type inequalities, see for example ^{2, 3, 4, 5, 6, 7, 8, 9}. Until now, a large number of research papers and books have been written on Ostrowski inequalities and their numerous applications.

The remainder of this work is organized as follows: In this section, the related definitions and theorems are summarised. In Section 2, new Ostrowski type integral inequalities are proved via fractional integral operators. At the end some conclusions and further directions of research are discussed in Section 3.

Let us consider a bi-dimensional interval in with and A function is said to be convex on if for all and it satisfies the following inequality:

A modification for convex function on was defined by Dragomir [dragomir], as follows:

A function is said to be convex on the co-ordinates on if the partial mappings and are convex where defined for all and

A formal definition for co-ordinated convex function may be stated as follows:

Definition 1. A function is called co-ordinated convex on for all and , if it satisfies the following inequality:

(1.1)

The mappingis a co-ordinated concave on if the inequality (1.1) holds in reversed direction for all and .

Note that every convex function is co-ordinated convex but the converse is not generally true (see, ¹⁰).

In ¹⁰, Dragomir proved the following inequality which is Hermite-Hadamard type inequality for co-ordinated convex functions on the rectangle from the plane

Theorem 2. Suppose that is co-ordinated convex, then we have the following inequalities:

(1.2)

The above inequalities are sharp. The inequalities in (1.2) hold in reverse direction if the mapping is a co-ordinated concave mapping.

In ¹¹, Raina defined the following results connected with the general class of fractional integral operators.

(1.3)

where the coefficients is a bounded sequence of positive real numbers and is the set of real numbers. With the help of (1.3), in ¹¹ and ¹², Raina and Agarwal et al. defined the following left-sided and right-sided fractional integral operators, respectively, as follows:

(1.4)

(1.5)

where and is such that the integrals on the right side exists.

It is easy to verify that and are bounded integral operators on , if

(1.6)

In fact, for we have

(1.7)

and

(1.8)

where

The importance of these operators stems indeed from their generality. Many useful fractional integral operators can be obtained by specializing the coefficient Here, we just point out that the classical Riemann-Liouville fractional integrals and of order defined by (see, ^{13, 14})

(1.9)

and

(1.10)

follow easily by setting

in (1.4) and (1.5), and the boundedness of (1.9) and (1.10) on is also inherited from (1.7) and (1.8), (see, ¹²). In ¹⁵, authors give new definitions related to fractional integral operators for two variables functions:

Definition 2. Le The fractional integral operators for two variables functions with

and defined by

and

Similar the above definition, we introduce the following integrals:

and

Now the main findings are given below.

2. Ostrowski Type Inequalities via Fractional Integral Operators

In this section, we outline some significant lemma, theorems and known properties of some special inequalities used throughout the remaining of the paper. For the theorems, we shall need the fractional integral operator version of the special identity for functions in two independent variables given below (Lemma l1). In accordance with this purpose we prove this special identity and have results which are extension of bivariate Ostrowski type inequalities with the help of fractional integral operator.

Throughout this section, ,, , and denote the following expressions:

and

Lemma 1. Let be a partial differentiable mapping on with If , then the following identity for fractional integral operators for function in two independent variables holds

for

Proof. Using the integration by parts for double integrals, we have

Similarly, following the same steps, we also have

and

Then by calculating we get the required identity.

In the light of this useful lemma, we can prove the following important theorems.

Theorem 3. Let be a partial differentiable mapping on with If i.e. is bounded on , then the following inequality for fractional integral operators for function in two independent variables holds

for

Proof. By taking modulus of Lemma 1, and using the property of boundedness of , we get

where and . Using the fact that

we have

which completes the proof.

Theorem 4. Let be a partial differentiable mapping on with . If , then the following inequality for fractional integral operators for function in two independent variables holds for

where

and , and are defined as above.

Proof. By taking modulus of Lemma 1 and applying the well-known Hölder's inequality for double integrals, we obtain

(2.1)

where and

Using the fact that

we obtain the required result.

Theorem 5. Let be a partial differentiable mapping on with If , then the following inequality for fractional integral operators for function in two independent variables holds for

where

Proof. Similarly by taking modulus of Lemma 1 and applying well-known power mean inequality for double integrals, we get

Then by rearranging the above inequality, we have

Similarly, we also have

and

Moreover, we get

and

Using the fact that

we obtain the desired inequality.

Theorem 6. Let be a partial differentiable mapping on with If is a co-ordinated concave mapping on , then the following inequality for fractional integral operators for function in two independent variables holds

(2.2)

where , , , , and are defined as above.

Proof. Since is a co-ordinated concave mapping on using the (1.2) with inequalities in reversed direction, we have

(2.3)

Similarly, we get

(2.4)

(2.5)

and

(2.6)

Using the inequalities (2.3)-(2.6) in (2.1), we obtain the required inequality (2.2). Thus, the proof is completed.

3. Concluding Remarks

In this study, first of all, we provide the practical identity for two independent variables with the help of fractional integral operators. Then by using this identity we present new upper bounds for bivariate Ostrowski type inequalities by taking advantage of co-ordinated concave mappings on closed intervals.

References