On Hadamard-type Inequalities for s-Preinvex Functions

In recent years,various refinements of the classical Hadamard inequalities for the convex functions and its variant forms are obtained in the literature by many researchers. At the same time,several refinements and variant forms of the Hadamard-type inequalities for s-convex functions as a generalization of convex functions,are also derived. The objective of this paper is to obtain several new Hadamard-type inequalities about s-preinvex functions. A new kind of generalized convex functions,termed s-preinvex functions in the second sense is introduced through relaxing the concept of s-convex functions. And the Hadamard-type inequalities for s-preinvex functions are established under certain conditions,i. e. let K[0,∞) be an invex set with respect to η. Assuming that f:K = [a,a + η(b,a) ]→[0,∞) is an s-preinvex function in K. a,b∈K,a a + η(b,a) ,then for some fixe s ∈(0,1],2s-1f((2a+η b,a/2)) ≤(1/ηb,a)) ≤(1/ηb,a)) ∫a+η(b,a) a f(x) dx ≤((f(a+) f(b)) /(a+1)) ,where η satisfies the well-known condition C:η(y,y + λη(x,y) ) =-λη(x,y);η(x,y + λη(x,y) ) =(1-λ) η(x,y) ,x,y ∈ R,λ ∈ [0,1]for the first inequality in the above inequalities. With Kirmaci's two new Hadamard-type inequalities for products of convex and s-convex functions,two new Hadamard-type inequalities for products of two s-preinvex functions are obtained. These results generalize some known results and include the previous known conclusions for s-convex as special case.