Hermite–Hadamard and Fejér Inequalities for Co-Ordinated ( F , G ) -Convex Functions on a Rectangle

: We introduce the notion of a co-ordinated ( F , G ) -convex function deﬁned on an interval in R 2 and we prove the Hermite–Hadamard and Fejér type inequalities for such functions.


Introduction
The celebrated inequality states that, if f : [a, b] → R is a convex function, then Furthermore, if p : [a, b] → [0, ∞) is an integrable function symmetric with respect to a+b 2 , that is then the following weighted generalization of the Hermite-Hadamard inequality is known as the Fejér inequality Dragomir [1] established a counterpart of the Hermite-Hadamard inequality for co-ordinated convex functions, that is functions f : [a, b] × [c, d] → R which are convex with respect to each variable separately.It has been proven in [1] that for such functions, the following inequalities hold Refinement versions of these inequalities have been presented in [1][2][3].
A counterpart of the Fejér inequality for co-ordinated convex functions has been formulated by Alomari and Darus [4].They proved that if p : [a, b] × [c, d] → [0, ∞) is an integrable function symmetric with respect to the lines x = a+b 2 and y = c+d 2 , i.e., p(a and then for every co-ordinated convex function the following inequalities hold In recent years, several modifications of the notion of convexity were studied by many authors (see e.g., [5][6][7][8][9]).The following general definition was introduced in [10].
In particular, if F is of the form where where C ∈ (0, ∞), then it is called strongly convex with modulus C. For some applications of F-convex functions in the optimization theory and in the theory of partial differential equations we refer to [11] and [12], respectively.It should be noted here that, although a definition of the F-convex function does not require any additional properties of F, it is reasonable to assume that F is symmetric, that is In fact, if f is F-convex then there exists a symmetric function F s such that f is F s -convex and To find this, one could take Note that F given by (4) or ( 5) is symmetric.Moreover, a symmetry of F is a necessary condition for the existence of an F-affine function, i.e., a function satisfying equation In what follows we deal with the functions of two variables, which are F-convex with respect to each variable.
Following the remark formulated above, we restrict our attention to the case where respectively.This assumption will not be repeated.Our main aim is to present the Hermite-Hadamard and the Fejér type inequalities for co-ordinated (F, G)-convex functions.

Hermite-Hadamard Type Inequalities
In this section, we prove the Hermite-Hadamard type inequalities for (F, G)-convex functions.Our proof is based on some methods used in [1] and [3].We begin with the result establishing the Hermite-Hadamard type inequalities for F-convex functions.It will be useful in further considerations. and which gives (8).Note also that, as f is F-convex, we have Setting in (9) x = ta + (1 − t)b, y = tb + (1 − t)a, where t ∈ [0, 1], and integrating obtained in this way inequality with respect to t, we obtain (7).Now, we are going to formulate and prove the Hermite-Hadamard type inequalities for co-ordinated (F, G)-convex functions. where where where where , y, c + d − y dy.
Integrating this inequality with respect to x, we find Moreover, since for every y ∈ [c, d], f (•, y) is F(y, •, •, •)-convex, using the similar arguments, we conclude that Adding up these inequalities, we obtain (11) and (12).

Fejér Type Inequalities
In order to prove the Fejér type inequalities for co-ordinated (F, G)-convex functions we need the following auxiliary result.
Proof.We prove only the first part of the lemma since the proof of the second part is similar.Assume for every y ∈ [c, d], we obtain In the next theorem we establish the Fejér type inequalities for (F, G)-convex functions. where where Therefore, as p is symmetric with respect to the lines x = a+b 2 and y = c+d 2 , we obtain Thus, (14) holds.Furthermore, using again the symmetry of p and applying Lemma 1 to [y, c which gives (15).

Discussion
In this paper the Hermite-Hadamard and Fejér type inequalities for co-ordinated (F, G)-convex functions are proved.Since every co-ordinated convex function is co-ordinated (F, G)-convex (with F and G being identically 0), from our results, one can easily deduce the results by Dragomir [1] and Alomari and Darus [4].Furthermore, applying Theorems 2 and 3, one can obtain the Hermite-Hadamard and Fejér type inequalities for co-ordinated (C, D)-approximately convex functions and co-ordinated (C, D)-strongly convex functions defined by Note also that from Theorem 1 the Hermite-Hadamard inequalities for approximately convex functions and strongly convex functions can be derived.Finally, applying Theorem 1, with F ≡ 0, we obtain the classical Hermite-Hadamard inequality.

Theorem 3 .
Assume that p : [a, b] × [c, d] → R is a positive integrable function symmetric with respect to the lines x = a+b 2 and y = c+d 2 (cf.(1) and (2) a + b − x p(x,