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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 Proof of One Optimal Inequality for | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 902432 5 pages doi 10.1155 2010 902432 Research Article Proof of One Optimal Inequality for Generalized Logarithmic Arithmetic and Geometric Means Ladislav Matejicka Faculty of Industrial Technologies in Púchov Alexander Dubcek University in TrenCín I. Krasku 491 30 02001 Puchov Slovakia Correspondence should be addressed to Ladislav Matejicka matejicka@tnuni.sk Received 11 July 2010 Revised 19 October 2010 Accepted 31 October 2010 Academic Editor Sin E. Takahasi Copyright 2010 Ladislav Matejicka. 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. Two open problems were posed in the work of Long and Chu 2010 . In this paper we give the solutions of these problems. 1. Introduction The arithmetic A a b and geometric G a b means of two positive numbers a and b are defined by A a b a b 2 G a b cOb respectively. If p is a real number then the generalized logarithmic mean Lp a b with parameter p of two positive numbers a b is defined by a bP 1 - aP 1 11 p p 1 b - a Lp a b - b - a ln b - ln a a b p 0 p - 1 a b p 0 a fb p -1 a fb. 1.1 In the paper 1 Long and Chu propose the two following open problems 2 Journal of Inequalities and Applications Open Problem 1. What is the least value p such that the inequality aA a b 1 - a G a b Lp a b 1.2 holds for a e 0 1 2 and all a b 0 with a fb Open Problem 2. What is the greatest value q such that the inequality aA a b 1 - a G a b Lq a b 1.3 holds for a e 1 2 1 and all a b 0 with a fb For information on the history background properties and applications of inequalities for generalized logarithmic arithmetic and geometric means please refer to 1-19 and related references there in. The aim of this article is to prove the following Theorem 2.1. 2. Main Result Theorem 2.1. Let a e 0 1 2 u