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 An Optimal Double Inequality for Means Wei-Mao Qian and Ning-Guo Zheng | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 578310 11 pages doi 2010 578310 Research Article An Optimal Double Inequality for Means Wei-Mao Qian and Ning-Guo Zheng Huzhou Broadcast and TV University Huzhou 313000 China Correspondence should be addressed to Wei-Mao Qian qwm661977@ Received 3 September 2010 Accepted 27 September 2010 Academic Editor Alberto Cabada Copyright 2010 . Qian and . Zheng. 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. For p e R the generalized logarithmic mean Lp a b arithmetic mean A a b and geometric mean G a b of two positive numbers a and b are defined by Lp a b a a b Lp a b ap 1 - bp 1 p 1 a - b 1 p p 0 p - 1 a ẹb Lp a b 1 e bb aa 1 b-a p 0 a b Lp a b b - a in b - ln a p -1 a b A a b a b 2 and G a b Vab respectively. In this paper we give an answer to the open problem for a e 0 1 what are the greatest value p and the least value q such that the double inequality Lp a b Ga a b A1-a a b Lq a b holds for all a b 0 1. Introduction For p e R the generalized logarithmic mean Lp a b of two positive numbers a and b is defined by a b Lp a b ap 1 - bp 1 11 p p 1 a - b 1 bb 1 b-a e aa b - a ln b - ln a p 0 p - 1 a b p 0 a b p -1 a b. It is wellknown that Lp a b is continuous and increasing with respect to p e R for fixed a and b. In the recent past the generalized logarithmic mean has been the subject of 2 Journal of Inequalities and Applications intensive research. Many remarkable inequalities and monotonicity results can be found in the literature 1-9 . It might be surprising that the generalized logarithmic mean has applications in physics economics and even in meteorology 10-13 . If we denote by A a b a b 2 I a b 1 e bb aa y d -b L a b b-a ln bln a G a b Vab and H a b 2ab a b the arithmetic mean identric
