On Some Integral Inequalities Related to Hermite-Hadamard-Fejér Inequalities for Coordinated Convex Functions

Several new mappings associated with coordinated convexity are proposed, by which we obtain some new Hermite-Hadamard-Fejér type inequalities for coordinated convex functions. We conclude that the results obtained in this work are the generalizations of the earlier results. 1. Introduction Let be a convex function and with ; then is known as the Hermite-Hadamard inequality. In [1], Fejér established the following weighted generalization of inequality (1). Theorem 1. If is a convex function, then the inequality holds, where is positive, integrable, and symmetric about . Inequalities (1) and (2) have been extended, generalized, and improved by a number of authors (e.g., [2–9]). In [4], Dragomir proposed the following Hermite-Hadamard type inequalities which refine the first inequality of (1). Theorem 2 (see [4]). Let be convex on . Then is convex, increasing on , and, for all , where An analogous result for convex functions which refines the second inequality of (1) is obtained by Yang and Hong in [10] as follows. Theorem 3 (see [10]). Let be convex on . Then is convex, increasing on , and, for all , where 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 4. A function is said to be convex on coordinates on if the inequality holds for all , , , and and . Dragomir in [5] established the following Hermite-Hadamard type inequalities for coordinated convex functions in a rectangle from the plane . Theorem 5. Suppose that is convex on the coordinates on . Then one has the following inequalities: The mapping connected with the first inequality of (9) is considered in [5]. If is a coordinated convex function, then the following mapping on can be defined by The mapping has the following properties:(i)is coordinated convex and monotonic nondecreasing on ;(ii)we have the following bounds for : Recently, Hwang et al. [11] established a monotonic nondecreasing mapping connected with the Hadamard’s inequality for coordinated convex functions in a rectangle from the plane as follows. Theorem 6 (see [11]). Suppose that is coordinated convex on and the mapping is defined by Then(i)the mapping is coordinated convex on ;(ii)the mapping is coordinated monotonic