Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 146945, 11 pages
doi:10.1155/2010/146945
Research Article

An Optimal Double Inequality between Power-Type Heron and Seiffert Means

Yu-Ming Chu,1 Miao-Kun Wang,2 and Ye-Fang Qiu2

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

Received 29 August 2010; Accepted 16 November 2010

Academic Editor: Alexander I. Domoshnitsky

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

For 𝑘 ∈ [ 0 , + ∞ ) , the power-type Heron mean 𝐻 𝑘 ( 𝑎 , 𝑏 ) and the Seiffert mean 𝑇 ( 𝑎 , 𝑏 ) of two positive real numbers 𝑎 and 𝑏 are defined by 𝐻 𝑘 ( 𝑎 , 𝑏 ) = ( ( 𝑎 𝑘 + ( 𝑎 𝑏 ) 𝑘 / 2 + 𝑏 𝑘 ) / 3 ) 1 / 𝑘 , 𝑘 ≠ 0 ; 𝐻 𝑘 √ ( 𝑎 , 𝑏 ) = 𝑎 𝑏 , 𝑘 = 0 and 𝑇 ( 𝑎 , 𝑏 ) = ( 𝑎 − 𝑏 ) / 2 a r c t a n ( ( 𝑎 − 𝑏 ) / ( 𝑎 + 𝑏 ) ) , 𝑎 ≠ 𝑏 ; 𝑇 ( 𝑎 , 𝑏 ) = 𝑎 , 𝑎 = 𝑏 , respectively. In this paper, we find the greatest value 𝑝 and the least value 𝑞 such that the double inequality 𝐻 𝑝 ( 𝑎 , 𝑏 ) < 𝑇 ( 𝑎 , 𝑏 ) < 𝐻 𝑞 ( 𝑎 , 𝑏 ) holds for all 𝑎 , 𝑏 > 0 with 𝑎 ≠ 𝑏 .