Generalizations of Inequalities for Differentiable Co-Ordinated Convex Functions

A generalized lemmas is proved and several new inequalities for differentiable co-ordinated convex and concave functions in two variables are obtained. 1. Introduction Let be a convex function and with ; we have the following double inequality: This remarkable result is well known in the literature as the Hermite-Hadamard inequality for convex mapping. Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1–4]). A modification for convex functions which is also known as coordinated convex functions was introduced as follows by Dragomir in [5]. Let us consider the bidimensional interval in with and ; a mapping is said to be convex on if the inequality holds for all , , and . A function is said to be coordinated convex on if the partial mappings , and , are convex for all and . A formal definition for coordinated convex functions may be stated as follows. Definition 1. A function is said to be convex on coordinates on if the inequality holds for all , , , , and . Dragomir in [5] established the following Hadamard-type inequalities for coordinated convex functions in a rectangle from the plane . Theorem 2. Suppose that is convex on the coordinates on . Then one has the inequalities as follows: Some new integral inequalities that are related to the Hermite-Hadamard type for coordinated convex functions are also established by many authors. In ([6], 2008), Alomari and Darus defined coordinated -convex functions and proved some inequalities based on this definition. In ([7], 2009), analogous results for -convex functions on the coordinates were proved by Latif and Alomari. In ([8], 2009), Alomari and Darus established some Hadamard-type inequalities for coordinated log-convex functions. In ([9], 2012), Latif and Dragomir obtained some new Hadamard type inequalities for differentiable coordinated convex and concave functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for coordinated convex functions in two variables based on the following lemma. Lemma 3. Let be a partial differentiable mapping on in with and . If , then the following equality holds: where Theorem 4 (see [9]). Let be a partial differentiable mapping on in with and . If is convex on the coordinates on , then the following equality holds: where Theorem 5 (see [9]). Let be a partial differentiable mapping on in with and . If is convex on the coordinates on and , , , then the following equality holds: where is as given in Theorem 4. Theorem 6 (see [9]).