Some New Inequalities of Hermite-Hadamard Type for ( )-( )-Convex Functions on Coordinates

The author introduces a new concept “( )-( )-convex functions on coordinates” and establishes some new inequalities of Hermite-Hadamard type for ( )-( )-convex functions of two variables on co-ordinates. 1. Introduction The following definitions are well known in the literature. Definition 1. Let be a function on an interval . If is valid for all and , then we say that is a convex function on . Definition 2 (see [1]). For and , if is valid for all and , then we say that is an -convex function on . Definition 3 (see [2]). For and , if is valid for all and , then we say that is an -convex function on . Definition 4 (see [3, 4]). For and , a function is said to be convex on coordinates, if the partial mappings are convex for all and . A formal definition for coordinated convex functions may be stated as follows. Definition 5. A function is said to be convex on coordinates, if holds for all , . Some inequalities of Hermite-Hadamard type for convex functions on coordinates may be recited as follows. Theorem 6 (see Theorem 1 in [3, 4]). Let be a convex function on coordinates. Then Theorem 7 (see [5, Theorem 2.3]). Let be a partial differentiable function. If is a convex function on coordinates, then where Theorem 8 (see [6, Theorem 2]). Let be a partial differentiable function. If is a convex function on coordinates, then where For more information on inequalities of Hermite-Hadamard type for various - -convex functions on coordinates and for extended -convex functions on coordinates, please refer to the recently published articles [7, 8] and related references therein. The aim of this paper is to introduce a new concept “ - -convex functions on coordinates” and to establish some new inequalities of Hermite-Hadamard type for - -convex functions of two variables on co-ordinates. 2. A Definition and a Lemma We now introduce a new notion “ - -convex functions on coordinates.” Definition 9. For some , a function is said to be - -convex on coordinates, if holds for all , . In order to establish some new inequalities of Hermite-Hadamard type for - -convex functions of two variables on coordinates, we need the following lemma. Lemma 10. Let be a partial differentiable function and denote the set of all Lebesgue integrable functions on . If , then Proof. Integrating by part yields Choosing and for and multiplying by on both sides of the above equations lead to the identity (12). The proof of Lemma 10 is complete. 3. Some Integral Inequalities of Hermite-Hadamard Type We are now in a position to establish new inequalities of Hermite-Hadamard type for - -convex