Abstract and Applied Analysis
Volume 2010 (2010), Article ID 303286, 13 pages
doi:10.1155/2010/303286
Research Article

Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means

Yu-Ming Chu1 and Bo-Yong Long2

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2College of Mathematics Science, Anhui University, Hefei 230039, China

Received 26 November 2009; Revised 16 February 2010; Accepted 16 March 2010

Academic Editor: Lance Littlejohn

Copyright © 2010 Yu-Ming Chu and Bo-Yong Long. 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.

Abstract

We answer the question: for α,β,γ∈(0,1) with α+β+γ=1, what are the greatest value p and the least value q, such that the double inequality Lp(a,b)<Aα(a,b)Gβ(a,b)Hγ(a,b)<Lq(a,b) holds for all a,b>0 with a≠b? Here Lp(a,b), A(a,b), G(a,b), and H(a,b) denote the generalized logarithmic, arithmetic, geometric, and harmonic means of two positive numbers a and b, respectively.