关于s-预不变凸函数的Hadamard型不等式  

On Hadamard-type Inequalities for s-Preinvex Functions

作　　者：李觉友[1] 

机构地区：[1]重庆师范大学数学学院,重庆400047

出　　处：《重庆师范大学学报（自然科学版）》2010年第4期5-8,共4页Journal of Chongqing Normal University:Natural Science

基　　金：国家自然科学基金(No.10671207);重庆师范大学青年基金资助项目(No.08XLQ03)

摘　　要：本文得到了关于s-预不变凸函数的3个Hadamard型不等式。首先通过推广s-凸函数的概念,定义了一类广义凸函数—s-预不变凸函数。同时使用推导s-凸函数的Hadamard不等式的类似方法,给出了s-预不变凸函数的Hadamard型不等式,即设K=[a,a+η(b,a)][0,∞)关于η为不变凸集,f:K→[0,∞)在K上为s-预不变凸函数,a,b∈K,a<a+η(b,a),则有2s-1f((2a+η b,a/2))≤(1/ηb,a))∫a+η(b,a)af(x)dx≤((f(a+)f(b))/(a+1)),其中上式第一个不等式中的η满足条件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]。最后还得到了有关两个s-预不变凸函数乘积形式的Hadamard型不等式。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 fixes ∈ （0,1 ],2s-1f（（2a＋η b,a/2））≤（1/ηb,a））∫a＋η（b,a）af（x）dx≤（（f（a＋）f（b））/（a＋1））where η satisfies the well-known conditionC：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 Kirrnaci＇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.

关 键 词：s-凸函数 s-预不变凸函数 HADAMARD型不等式 

分 类 号：O174.13[理学—数学]