NEW INEQUALITIES OF HERMITE-HADAMARD AND FEJÉR TYPE INEQUALITIES VIA PREINVEXITY

Several new weighted inequalities connected with Hermite-Hadamard and Fejér type inequalities are established for functions whose derivatives in absolute value are preinvex. The results presented in this paper provide extensions of those given in earlier works. 1. Introduction Many inequalities have been established for convex functions but the most famous is the Hermite-Hadamard inequality, due to its rich geometrical signi cance and applications, which is stated as (see [23]): Let f : I R! R be a convex mapping 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 : Both the inequalities hold in reversed direction if f is concave. In [7], Fejér gave a weighted generalization of (1.1) as follows: (1.2) f a+ b 2 Z b a w(x)dx Z b a f(x)w(x)dx f(a) + f(b) 2 Z b a w(x)dx; where f : [a; b] ! R be a convex function and f : [a; b] ! R is nonnegative, integrable and symmetric about x = a+b 2 . For several results which generalize, improve and extend the inequalities (1.1) and (1.2) we refer the interested reader [5, 6, 8], [10]-[13], [23, 24], [26]-[31]. In recent years, lot of e¤orts have been made by many mathematicians to generalize the classical convexity. These studies include among others the work of Hanson in [9], Ben-Israel and Mond [4], Pini [21], M.A.Noor [18, 19], Yang and Li [33] and Weir [32]. Mond [5], Weir [32] and Noor [18, 19], have studied the basic properties of the preinvex functions and their role in optimization, variational inequalities and equilibrium problems. Hanson in [9], introduced invex functions as a signi cant generalization of convex functions. Ben-Israel and Mond [4], gave the concept of preinvex function which is special case of invexity. Pini [21], introduced the concept of prequasiinvex functions as a generalization of invex functions. Let us recall some known results concerning invexity and preinvexity Date : Today. 2000 Mathematics Subject Classi cation. 26D15, 26D20, 26D07. Key words and phrases. Hermite-Hadamards inequality,Fejérs inequality, invex set, preinvex function, Hölders integral inequality. This paper is in nal form and no version of it will be submitted for publication elsewhere. 1 2 M. A. LATIF Let K be a closed set in R and let f : K ! R and : K K ! R be continuous functions. Let x 2 K, then the set K is said to be invex at x with respect to ( ; ), if x+ t (y; x) 2 K;8x; y 2 K; t 2 [0; 1]: K is said to be an invex set with respect to if K is invex at each x 2 K. The invex set K is also called a -connected set. De nition 1. [32]The function f on the invex set K is said to be preinvex with respect to , if f(u+ t (v; u)) (1 t) f(u) + tf(v);8u; v 2 K; t 2 [0; 1]: The function f is said to be preconcave if and only if f is preinvex. It is to be noted that every convex function is preinvex with respect to the map (x; y) = x y but the converse is not true see for instance [32]. In the recent paper, Noor [16] has obtained the following Hermite-Hadamard inequalities for the preinvex functions: Theorem 1. [16]Let f : [a; a+ (b; a)] ! (0;1) be a preinvex function on the interval of the real numbers K (the interior of K) and a, b 2 K with a < a+ (b; a). Then the following inequality holds: (1.3) f 2a+ (b; a) 2 1 (b; a) Z a+ (b;a) a f (x) dx f (a) + f (b) 2 : For several new results on inequalities for preinvex functions we refer the interested reader to [3, 15, 20, 25] and the references therein. In the present paper we give new inequalities of Hermite-Hadamard and Fejér for functions whose derivatives are preinvex. Our results generalize those results presented in a very recent paper of M. Z. Sarikaya [25, 28]. 2. Main Results The following Lemma is essential in establishing our main results in this section: Lemma 1. Let K R be an open invex subset with respect to : K K ! R and a, b 2 K with a < a+ (b; a) Suppose f : K ! R is a di¤erentiable mapping on K such that f 0 2 L ([a; a+ (b; a)]). If w : [a; a+ (b; a)] ! [0;1) is an integrable mapping, then for every a; b 2 K with (b; a) 6= 0 the following equality holds: (2.1) 1 (b; a) Z a+ (b;a) a f (x)w(x)dx 1 (b; a) f a+ 1 2 (b; a) Z a+ (b;a) a w(x)dx