Journal of Applied Mathematics
Volume 2011 (2011), Article ID 261237, 6 pages
http://dx.doi.org/10.1155/2011/261237
Research Article

An Optimal Double Inequality between Seiffert and Geometric Means

Yu-Ming Chu,1 Miao-Kun Wang,1 and Zi-Kui Wang2

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

Received 30 June 2011; Accepted 14 October 2011

Academic Editor: J. C. Butcher

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

For 𝛼 , 𝛽 ∈ ( 0 , 1 / 2 ) we prove that the double inequality 𝐺 ( 𝛼 𝑎 + ( 1 − 𝛼 ) 𝑏 , 𝛼 𝑏 + ( 1 − 𝛼 ) 𝑎 ) < 𝑃 ( 𝑎 , 𝑏 ) < 𝐺 ( 𝛽 𝑎 + ( 1 − 𝛽 ) 𝑏 , 𝛽 𝑏 + ( 1 − 𝛽 ) 𝑎 ) holds for all 𝑎 , 𝑏 > 0 with 𝑎 ≠ 𝑏 if and only if √ 𝛼 ≤ ( 1 − 1 − 4 / 𝜋 2 ) / 2 and √ 𝛽 ≥ ( 3 − 3 ) / 6 . Here, 𝐺 ( 𝑎 , 𝑏 ) and 𝑃 ( 𝑎 , 𝑏 ) denote the geometric and Seiffert means of two positive numbers 𝑎 and 𝑏 , respectively.