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

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

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.