Abstract and Applied Analysis
Volume 2010 (2010), Article ID 604804, 9 pages
doi:10.1155/2010/604804
Research Article

The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

Wei-Feng Xia,1 Yu-Ming Chu,2 and Gen-Di Wang2

1School of Teacher Education, Huzhou Teachers College, Huzhou, Zhejiang 313000, China
2Department of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang 313000, China

Received 16 December 2009; Accepted 12 March 2010

Academic Editor: Lance Littlejohn

Copyright © 2010 Wei-Feng Xia 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

For p∈ℝ, the power mean Mp(a,b) of order p, logarithmic mean L(a,b), and arithmetic mean A(a,b) of two positive real values a and b are defined by Mp(a,b)=((ap+bp)/2)1/p, for p≠0 and Mp(a,b)=ab, for p=0, L(a,b)=(b-a)/(log⁡b-log⁡a), for a≠b and L(a,b)=a, for a=b and A(a,b)=(a+b)/2, respectively. In this paper, we answer the question: for α∈(0,1), what are the greatest value p and the least value q, such that the double inequality Mp(a,b)≤αA(a,b)+(1-α)L(a,b)≤Mq(a,b) holds for all a,b>0?