Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 436457, 7 pages
doi:10.1155/2010/436457
Research Article

The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

Yu-Ming Chu,1 Ye-Fang Qiu,2 Miao-Kun Wang,2 and Gen-Di Wang1

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received 28 December 2009; Revised 17 April 2010; Accepted 22 April 2010

Academic Editor: Andrea Laforgia

Copyright © 2010 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 greatest value α and least value β such that the double inequality αA(a,b)+(1-α)H(a,b)<P(a,b)<βA(a,b)+(1-β)H(a,b) holds for all a,b>0 with a≠b. Here A(a,b), H(a,b), and P(a,b) denote the arithmetic, harmonic, and Seiffert's means of two positive numbers a and b, respectively.