# On the Hadamard Type Inequalities Involving Product of Two Convex Functions on the Co-Ordinates

###### Abstract.

In this paper some Hadamard-type inequalities for product of convex funcitons of variables on the co-ordinates are given.

###### Key words and phrases:

Convex, convex, Co-ordinates, Product of functions, Hadamard’s inequality, Beta function, Gamma function.Corresponding author

###### 1991 Mathematics Subject Classification:

Primary 26D15, Secondary 26A51## 1. Introduction

The inequality

(1.1) |

where is a convex function defined on the interval of the set of real numbers, and with is well known in the literature as Hadamard’s inequality.

For some recent results related to this classic inequality, see [1], [8], [11], [12], and [14], where further references are given.

In [2], Hudzik and Maligranda considered, among others, the class of functions which are convex in the second sense. This class is defined as following:

###### Definition 1.

A function is said to be convex in the second sense if

holds for all and for some fixed

The class of convex functions in the second sense is usually denoted with It is clear that if we choose we have ordinary convexity of functions defined on

In [15], Kırmacı et al., proved the following inequalities related to product of convex functions. These are given in the next theorems.

###### Theorem 1.

Let be functions such that and are in If is convex and nonnegative on and if is convex on for some fixed then

(1.2) |

where

###### Theorem 2.

Let be functions such that and are in If is convex and is convex on for some fixed then

(1.3) | |||||

###### Theorem 3.

Let be functions such that and are in If is convex and nonnegative on and if is convex on for some fixed then

In [12], Dragomir defined convex functions on the co-ordinates as follows and proved Lemma 1 related to this definiton:

###### Definition 2.

Let us consider the bidimensional interval in with A function will be called convex on the co-ordinates if the partial mappings and are convex where defined for all and Recall that the mapping is convex on if the following inequality holds,

for all and

###### Lemma 1.

Every convex mapping is convex on the co-ordinates, but converse is not general true.

A formal definition for co-ordinated convex functions may be stated as follow [see [16]]:

###### Definition 3.

A function is said to be convex on the co-ordinates on if the following inequality:

holds for all and .

In [12], Dragomir established the following inequalities:

###### Theorem 4.

Suppose that is convex on the co-ordinates on Then one has the inequalities:

In [7], Alomari and Darus defined convexity on as follows:

###### Definition 4.

Consider the bidimensional interval in with and The mapping is convex on if

holds for all with and for some fixed

In [7], Alomari and Darus proved the following lemma:

###### Lemma 2.

Every convex mappings is convex on the co-ordinates, but converse is not true in general .

In [4], Latif and Alomari established Hadamard-type inequalities for product of two convex functions on the co-ordinates as follow:

###### Theorem 5.

Let be convex functions on the co-ordinates on with and Then

where

###### Theorem 6.

## 2. Main Results

###### Theorem 7.

Let be convex function on the co-ordinates and be convex function on the co-ordinates with and for some fixed Then one has the inequality:

where

###### Proof.

Since is co-ordinated convex and is co-ordinated convex, from Lemma 1 and Lemma 2, the partial mappings

are convex on and respectively, where . Similarly;

are convex on and respectively, where

In the next theorem we will also make use of the Beta function of Euler type, which is for defined as

and the Gamma function is defined as

###### Theorem 8.

Let be convex function on the co-ordinates and be convex functions on the co-ordinates with and for some fixed Then one has the inequality:

where

###### Proof.

Since is co-ordinated convex and is co-ordinated convex, from Lemma 2, the partial mappings

are convex on and respectively, where Similarly;

are convex on and respectively, where

###### Theorem 9.

Let be convex function on the co-ordinates and be convex function on the co-ordinates with and for some fixed Then one has the inequality: