GENERALIZED HERMITE-HADAMARD TYPE INEQUALITIES FOR PRODUCTS OF CO-ORDINATED CONVEX FUNCTIONS

where f : I ⊂ R→ R is a convex function on the interval I and a, b ∈ I with a < b. Hermite—Hadamard inequality provides a lower and an upper estimation for the integral average of any convex function defined on a compact interval. This inequality has a notable place in mathematical analysis, optimization and so on. However, many studies have been established to demonstrate its new proofs, refinements, extensions and generalizations. A few of these studies are ([4], [9]-[11], [13]-[17], [24]-[27], [29], [34], [35], [37]) referenced works and also the references included there. On the other hand, Hermite-Hadamard inequality is considered for convex functions on the co-ordinates in [12], [18]. If we look at the convexity of the co-ordinates, there are a lot of definitions of co-ordinated convex function. They may be stated as follows [12]:


Introduction
The following inequality discovered by C. Hermite and J. Hadamard for convex functions is well known in the literature as the Hermite-Hadamard inequality (see, e.g., [13]): where f : I R ! R is a convex function on the interval I and a; b 2 I with a < b: Hermite-Hadamard inequality provides a lower and an upper estimation for the integral average of any convex function de…ned on a compact interval. This inequality has a notable place in mathematical analysis, optimization and so on. However, many studies have been established to demonstrate its new proofs, re…nements, extensions and generalizations. A few of these studies are ( [4], [9]- [11], [13]- [17], [24]- [27], [29], [34], [35], [37]) referenced works and also the references included there.
On the other hand, Hermite-Hadamard inequality is considered for convex functions on the co-ordinates in [12], [18]. If we look at the convexity of the co-ordinates, there are a lot of de…nitions of co-ordinated convex function. They may be stated as follows [12]: 864 HÜSEYIN BUDAK AND TUBA TUNÇ De…nition 1. Let us consider a bidimensional interval := [a; b] [c; d] in R 2 with a < b and c < d: A function f : R 2 ! R is said to be convex on if the following inequality satis…es for all (x; y); (z; w) 2 and t 2 [0; 1].
A modi…cation of de…nition of co-ordinated convex function was de…ned by Dragomir [12] as follows: ! R is said to be convex on the co-ordinates on if the partial mappings f y : A formal de…nition for co-ordinated convex function may be stated as follows: The following Hermite-Hadamard type inequalities for co-ordinated convex functions were obtained by Dragomir in [12]: ! R is co-ordinated convex, then we have the following inequalities: b The following Hermite-Hadamard type inequality utilizing co-ordinated convex functions was proved by Sarikaya in [28]: with 0 a < b and 0 c < d and f 2 L( ). Then for ; > 0 we have the following Hermite-Hadamard type inequality Now, let's give the notations A k (x; m; n) and B k (x; m; n) used throughout the study: In [7], Budak gave the following inequalities which are used the main results: Theorem 8. Suppose that w 1 : [a; b] ! R is non-negative, integrable and symmetric about x = a+b 2 (i.e. w 1 (x) = w 1 (a + b x)). If f; g : I ! R are two real-valued, non-negative and convex functions on I, then for any a; b 2 Theorem 9. Suppose that conditions of Theorem 8 hold, then we have the following inequality where M (a; b) and N (a; b) are de…ned as in Theorem 8.
The aim of this paper is to establish Hermite-Hadamard type inequalities for product of co-ordinated convex functions. The results presented in this paper provide extensions of those given in [6] and [18]
Proof. Since f and g are co-ordinated convex functions on , the functions f x and g x are convex on [c; d]. If the inequality (6) is applied for the functions f x and g x , then we obtain That is, Multiplying the inequality (9) by w1(x) (b a) and then integrating respect to x from a to b; we get Applying the inequality (6) to each integrals in (10) Substituting the inequalities (11)- (14) in the inequality (10) and then arranging the result obtained, we get desired result. On the other hand, the same result is obtained by using the convexity of functions f y and g y .
Theorem 11. Let f; g : R 2 ! [0; 1) be co-ordinated convex functions on with a < b; c < d. Also, w 1 : [a; b] ! R is non-negative, integrable and symmetric about x = a+b 2 (i.e. w 1 (x) = w 1 (a + b x)) and w 2 : [c; d] ! R is non-negative, integrable and symmetric about y = c+d 2 (i.e. w 2 (y) = w 2 (c + d y)). Then, we have the following Hermite-Hadamard type inequality Proof. Since f and g are co-ordinated convex functions on , the functions f x , g x , f y and g y are convex. Applying the inequality (7) for the functions f c+d we get Similarly, if we apply the inequality (7) for the functions f a+b B 2 (y; c; d) A 2 (y; c; d): Using the inequality (7) for each integrals in inequalities (15) and (16), we have When the inequalities (17)-(25) is written in (15) and (16) and then the results obtained are added side by side and rearranged, we obtain The inequality (7) is applied to f x; c+d 2 g x; c+d 2 and then the result is multiplied by w 1 (x) and integrated over [a; b];we get Similarly, if we apply the inequality (7) to f a+b 2 ; y g a+b 2 ; y and then the result is multiplied by w 2 (y) and integrated over [c; d];we get [f (a; y)g(b; y) + f (b; y)g(a; y)] w 2 (y)dy: Substituting the inequalities (27) and (28) in the inequality (26) and reordering the results obtained, we have [f (a; y)g(b; y) + f (b; y)g(a; y)] w 2 (y)dy By applying the inequality (6) to each integral in (29) and later rearranging the results obtained, we obtain desired inequality. which is proved by Budak and Sarikaya [6].