tailieunhanh - Báo cáo hóa học: " Research Article An Optimal Double Inequality between Power-Type Heron and Seiffert Means"

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 between Power-Type Heron and Seiffert Means | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 146945 11 pages doi 2010 146945 Research Article An Optimal Double Inequality between Power-Type Heron and Seiffert Means Yu-Ming Chu 1 Miao-Kun Wang 2 and Ye-Fang Qiu2 1 Department of Mathematics Huzhou Teachers College Huzhou 313000 China 2 Department of Mathematics Zhejiang Sci-Tech University Hangzhou 310018 China Correspondence should be addressed to Yu-Ming Chu chuyuming2005@ Received 29 August 2010 Accepted 16 November 2010 Academic Editor Alexander I. Domoshnitsky Copyright 2010 Yu-Ming Chu et al. 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 k e 0 x the power-type Heron mean Hk a b and the Seiffert mean T a b of two positive real numbers a and b are defined by Hk a b ak ab k 2 bk 3 1 k k 0 Hk a b Vab k 0 and T a b a - b 2arctan a - b a b a b T a b a a b respectively. In this paper we find the greatest value p and the least value q such that the double inequality Hp a b T a b Hq a b holds for all a b 0 with a fb. 1. Introduction For k e 0 to the power-type Heron mean Hk a b and the Seiffert mean T a b of two positive real numbers a and b are defined by Hk a b ak ab k 2 bk 1 k 3 k 0 T a b Vãb k 0 a - b - 2arctan a - b a b a a fb a b respectively. 2 Journal of Inequalities and Applications Recently the means of two variables have been the subject of intensive research 115 . In particular many remarkable inequalities for Hk a b and T a b can be found in the literature 16-20 . It is well known that Hk a b is continuous and strictly increasing with respect to k e 0 x for fixed a b 0 with a fb. Let A a b a b 2 1 a b 1 e bb aa 1 b a L a b b - a log b - log a G a b Vab and H a b 2ab a b be the arithmetic identric logarithmic geometric and harmonic means of two positive numbers a