tailieunhanh - Báo cáo sinh học: " Research Article Multiplicative Concavity of the Integral of Multiplicatively Concave Functions"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Multiplicative Concavity of the Integral of Multiplicatively Concave Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 845390 8 pages doi 2010 845390 Research Article Multiplicative Concavity of the Integral of Multiplicatively Concave Functions Yu-Ming Chu1 and Xiao-Ming Zhang2 1 Department of Mathematics Huzhou Teachers College Huzhou Zhejiang 313000 China 2 Haining College Zhejiang TV University Haining Zhejiang 314400 China Correspondence should be addressed to Yu-Ming Chu chuyuming2005@ Received 25 March 2010 Accepted 7 June 2010 Academic Editor S. S. Dragomir Copyright 2010 . Chu and . Zhang. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We prove that G x y ịy f t dtỊ is multiplicatively concave on a b X a b if f a b c 0 to 0 to is continuous and multiplicatively concave. 1. Introduction For convenience of the readers we first recall some definitions and notations as follows. Definition . Let I c R be an interval. A real-valued function f I R is said to be convex if f x Ay 2 for all x y e I. And f is called concave if -f is convex. Definition . Let I c 0 to be an interval. A real-valued function f I 0 to is said to be multiplicatively convex if f Vxy y f xf y for all x y e I. And f is called multiplicatively concave if 1 f is multiplicatively convex. 2 Journal of Inequalities and Applications For x x1 x2 6 Rị xi x2 x1 0 x2 0 and a 0 we denote log X log Xi log X2 xa xa xa 1-3 ex exi ex- . 1-4 For x x1 x2 y y1 y2 6 R2 we denote xy x1y1 x2y2 . 1-5 Definition . A set E1 Q R2 is said to be convex if x y 2 6 E1 whenever x y 6 E1- And a set E2 Q R2 is said to be multiplicatively convex if x1 2y1 2 6 E2 whenever x y 6 E2- From Definition 1-3 we clearly see that E1 Q Rị is a multiplicatively convex set if and only if log E1 logx x 6 E1 is a convex set and E2 Q R2 is a convex set if and