Abstract and Applied Analysis
Volume 2012 (2012), Article ID 425175, 6 pages
http://dx.doi.org/10.1155/2012/425175
Research Article

Sharp Bounds for Seiffert Mean in Terms of Contraharmonic Mean

Yu-Ming Chu1 and Shou-Wei Hou2

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

Received 17 September 2011; Accepted 5 November 2011

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Yu-Ming Chu and Shou-Wei Hou. 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 the least value 𝛽 in ( 1 / 2 , 1 ) such that the double inequality 𝐶 ( 𝛼 𝑎 + ( 1 − 𝛼 ) 𝑏 , 𝛼 𝑏 + ( 1 − 𝛼 ) 𝑎 ) < 𝑇 ( 𝑎 , 𝑏 ) < 𝐶 ( 𝛽 𝑎 + ( 1 − 𝛽 ) 𝑏 , 𝛽 𝑏 + ( 1 − 𝛽 ) 𝑎 ) holds for all 𝑎 , 𝑏 > 0 with 𝑎 ≠ 𝑏 . Here, 𝑇 ( 𝑎 , 𝑏 ) = ( 𝑎 − 𝑏 ) / [ 2 arctan ( ( 𝑎 − 𝑏 ) / ( 𝑎 + 𝑏 ) ) ] and 𝐶 ( 𝑎 , 𝑏 ) = ( 𝑎 2 + 𝑏 2 ) / ( 𝑎 + 𝑏 ) are the Seiffert and contraharmonic means of 𝑎 and 𝑏 , respectively.