Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 905679, 6 pages
doi:10.1155/2010/905679
Research Article

Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means

Bo-Yong Long1,2 and Yu-Ming Chu3

1College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
2School of Mathematical Sciences, Anhui University, Hefei 230039, China
3Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 13 November 2009; Accepted 25 February 2010

Academic Editor: Andrea Laforgia

Copyright © 2010 Bo-Yong Long and Yu-Ming Chu. 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 of order p of two positive numbers a and b is defined by Mp(a,b)=((ap+bp)/2)1/p, for p≠0, and Mp(a,b)=ab, for p=0. In this paper, we answer the question: what are the greatest value p and the least value q such that the double inequality Mp(a,b)≤Aα(a,b)Gβ(a,b)H1-α-β(a,b)≤Mq(a,b) holds for all a,b>0 and α,β>0 with α+β<1? Here A(a,b)=(a+b)/2, G(a,b)=ab, and H(a,b)=2ab/(a+b) denote the classical arithmetic, geometric, and harmonic means, respectively.