HADAMARD TYPE INEQUALITIES FOR ’CONVEX FUNCTIONS ON THE CO-ORDINATES ERHAN SET, MEHMET ZEKI SARIKAYA, AND AHMET OCAK AKDEMIR Abstract. In this paper, we introduce the notation of ’convex functions on the co-ordinates and present some properties. Also, new Hadamard type inequalities for such functions are obtained. 1. Introduction Let us recall some known denitions and results which we will use in this paper. If f is a convex function on the interval I =[a; b] and a; b 2 I with a<b, then (1.1) f  a + b 2   1 b  a Z b a f (x)dx  f (a)+ f (b) 2 which is known as the Hadamard inequality in the literature. (see, e.g., [4], [8, p.137]) If the function f is concave, reversed signs of inequality hold in (1.1). The inequality (1.1) is one of the most seful inequalities in mathematical analysis. For new proofs, numerious generalizatons, variants and extensions on this inequality, see ([3]-[8]) where further references are given. Let us consider a function ’ :[a; b] ! [a; b] where [a; b]  R. A function f :[a; b] ! R is said to be ’convex on [a; b] if for every two points x; y 2 [a; b] and t 2 [0; 1] the following inequality holds: f (t’(x) + (1  t)’(y))  tf (’(x)) + (1  t)f (’(y)) (see [9]) In [2], Cristescu established the following results for the ’convex functions. Lemma 1. For f :(a; b) ! R, the following statements are equivalent: (i) f is ’convex functions on [a; b] (ii) for every x; y 2 [a; b] ; the mapping g : [0; 1] ! R, g(t)= f (t’(x)+(1t)’(y)) is classically convex on [0; 1] : Obviously, if function ’ is the identity, then the classical convexity is obtained from the previous denition. For many properties of the ’convex functions, see [1],[2],[9]. In [2], Cristescu proved the following inequalities for ’convex functions: Theorem 1. If f :[a; b] ! R is ’convex for the continuous function ’ :[a; b] ! [a; b] then (1.2) f  ’ (a)+ ’ (b) 2   1 ’ (b)  ’ (a) Z ’(b) ’(a) f (x)dx  f (’ (a)) + f (’ (b)) 2 : 2000 Mathematics Subject Classication. 26A51, 26D07, 26D10, 26D15. Key words and phrases. Hadamard type inequalities, co-ordinated ’convex functions. 1