Abstract and Applied Analysis
Volume 2011 (2011), Article ID 520648, 9 pages
http://dx.doi.org/10.1155/2011/520648
Research Article

Optimal Lower Power Mean Bound for the Convex Combination of Harmonic and Logarithmic Means

Yu-Ming Chu,1 Shan-Shan Wang,2 and Cheng Zong2

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2School of Science, Hangzhou Normal University, Hangzhou 310012, China

Received 1 February 2011; Accepted 15 May 2011

Academic Editor: Irena Lasiecka

Copyright © 2011 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.

Abstract

We find the least value λ∈(0,1) and the greatest value p=p(α) such that αH(a,b)+(1−α)L(a,b)>Mp(a,b) for α∈[λ,1) and all a,b>0 with a≠b, where H(a,b), L(a,b), and Mp(a,b) are the harmonic, logarithmic, and p-th power means of two positive numbers a and b, respectively.