HERMITE-HADAMARD-FEJÉR INEQUALITIES FOR DOUBLE INTEGRALS

Both inequalities hold in the reversed direction if f is concave. We note that Hermite-Hadamard inequality may be regarded as a renement of the concept of convexity and it follows easily from Jensens inequality. Hermite-Hadamard inequality for convex functions has received renewed attention in recent years and a remarkable variety of renements and generalizations have been studied (see, for example, [4], [9][11], [14], [21], [25], [29], [30], [32]). The most well-known inequalities related to the integral mean of a convex function are the Hermite Hadamard inequalities or its weighted versions, the so-called


Introduction
The Hermite-Hadamard inequality discovered by C. Hermite and J. Hadamard (see, e.g., [8], [22, p.137]) is one of the most well established inequalities in the theory of convex functions with a geometrical interpretation and many applications. These inequalities state that if f : I ! R is a convex function on the interval I of real numbers and a; b 2 I with a < b, then Both inequalities hold in the reversed direction if f is concave. We note that Hermite-Hadamard inequality may be regarded as a re…nement of the concept of convexity and it follows easily from Jensen's inequality. Hermite-Hadamard inequality for convex functions has received renewed attention in recent years and a remarkable variety of re…nements and generalizations have been studied (see, for example, [4], [9]- [11], [14], [21], [25], [29], [30], [32]). The most well-known inequalities related to the integral mean of a convex function are the Hermite Hadamard inequalities or its weighted versions, the so-called Hermite-Hadamard-Fejér inequalities, In [13], Fejer gave a weighted generalization of the inequalities (1) as the following: Theorem 1. f : [a; b] ! R, be a convex function, then the inequality holds, where g : [a; b] ! R is nonnegative, integrable, and symmetric about x = a+b 2 (i.e. g(x) = g(a + b x)): A formal de…nation for co-ordinated convex function may be stated as follows: De…nition 2. A function f : ! R is called co-ordinated convex on ; for all (x; u); (y; v) 2 and t; s 2 [0; 1], if it sati…es the following inequality: The mapping f is a co-ordinated concave on if the inequality (3) holds in reversed direction for all t; s 2 [0; 1] and (x; u); (y; v) 2 .
The aim of this paper is to establish a new weighed generalizations of Hermite-Hadamard type integral inequalities (4). The results presented in this paper provide extensions of those given in [2], [7] and [12].
We will use the following lemma to proof of main result: Then for the convex partial mappings f y : the following hold: and

Hermite-Hadamard-Fejer Inequalities
Lets start the following Hermite-Hadamard-Fejer inequalities:  (2) for the convex function f y (x); then we obtain the inequality i.e.
Integrating the inequality (12) Summing the inequalities (10) and (13), we obtain the second and third inequalities in (7). Since f a+b 2 ; y is convex on [c; d] and p x (y) is positive, integrable and symmetric about c+d 2 ; using the …rst inequality in (2), we have Integrating the inequality (12) with respect to x on [a; b] ; we get Since f x; c+d 2 is convex on [c; d] and p y (x) is positive, integrable and symmetric about a+b 2 ; utilizing the …rst inequality in (2), we have the following inequality From the inequalities (15) and (16), we have the …rst inequality in (7).
For the proof of last inequality in (7), using the second inequality in (2) for the convex functions f (x; c) and f (x; d) on [a; b] and for the symmetric function p y (x), and Similarly, applying the second inequality in (2) for the convex functions f (a; y) and f (b; y) on [c; d] and for the symmetric function p x (y); we have and Integrating the inequalities (17) and (18) (19) and (20) on [a; b] ; then summing the resulting inequality we obtain the last inequality in (7). This completes the proof.

Remark 7.
Under assumptions of Theorem 6 with p(x; y) = 1; the inequalities (7) reduce to inequalities (4) proved by Dragomir in [7]. for all (x; y) 2 in Theorem 6, then we have the following inequalities which is the same result proved by Farid et al. in [12].