Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On The Hadamard’s Inequality for Log-Convex Functions on the Coordina | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 283147 13 pages doi 2009 283147 Research Article On The Hadamard s Inequality for Log-Convex Functions on the Coordinates Mohammad Alomari and Maslina Darus School of Mathematical Sciences Universiti Kebangsaan Malaysia UKM Bangi 43600 Selangor Malaysia Correspondence should be addressed to Maslina Darus maslina@ Received 15 January 2009 Revised 31 May 2009 Accepted 20 July 2009 Recommended by Sever Silvestru Dragomir Inequalities of the Hadamard and Jensen types for coordinated log-convex functions defined in a rectangle from the plane and other related results are given. Copyright 2009 M. Alomari and M. Darus. 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. 1. Introduction Let f I c R R be a convex mapping defined on the interval I of real numbers and a b I with a b then A a b a f b A f x dx 2----- L1 holds this inequality is known as the Hermite-Hadamard inequality. For refinements counterparts generalizations and new Hadamard-type inequalities see 1-8 . A positive function f is called log-convex on a real interval I a b if for all x y a b and X 0 1 f Xx 1 - X y fX x f 1-X y . If f is a positive log-concave function then the inequality is reversed. Equivalently a function f is log-convex on I if f is positive and log f is convex on I. Also if f 0 and f exists on I then f is log-convex if and only if f f - f 2 0. 2 Journal of Inequalities and Applications The logarithmic mean of the positive real numbers a b a b is defined as M _ b - a L a b L log b - log a A version of Hadamard s inequality for log-convex concave functions was given in 9 as follows. Theorem . Suppose that f is a positive log-convex function on a b then T-1 bf x dx Lf a f by b a a If f is a positive log-concave .
