Journal of Inequalities and Applications
Volume 2011 (2011), Article ID 686834, 9 pages
doi:10.1155/2011/686834
Research Article

The Optimal Convex Combination Bounds for Seiffert's Mean

Hong Liu1 and Xiang-Ju Meng2

1College of Mathematics and Computer Science, Hebei University, Baoding 071002, China
2Department of Mathematics, Baoding College, Baoding 071002, China

Received 28 November 2010; Accepted 28 February 2011

Academic Editor: P. Y. H. Pang

Copyright © 2011 Hong Liu and Xiang-Ju Meng. 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 derive some optimal convex combination bounds related to Seiffert's mean. We find the greatest values 𝛼 1 , 𝛼 2 and the least values 𝛽 1 , 𝛽 2 such that the double inequalities 𝛼 1 𝐶 ( 𝑎 , 𝑏 ) + ( 1 − 𝛼 1 ) 𝐺 ( 𝑎 , 𝑏 ) < 𝑃 ( 𝑎 , 𝑏 ) < 𝛽 1 𝐶 ( 𝑎 , 𝑏 ) + ( 1 − 𝛽 1 ) 𝐺 ( 𝑎 , 𝑏 ) and 𝛼 2 𝐶 ( 𝑎 , 𝑏 ) + ( 1 − 𝛼 2 ) 𝐻 ( 𝑎 , 𝑏 ) < 𝑃 ( 𝑎 , 𝑏 ) < 𝛽 2 𝐶 ( 𝑎 , 𝑏 ) + ( 1 − 𝛽 2 ) 𝐻 ( 𝑎 , 𝑏 ) hold for all 𝑎 , 𝑏 > 0 with 𝑎 ≠ 𝑏 . Here, 𝐶 ( 𝑎 , 𝑏 ) , 𝐺 ( 𝑎 , 𝑏 ) , 𝐻 ( 𝑎 , 𝑏 ) , and 𝑃 ( 𝑎 , 𝑏 ) denote the contraharmonic, geometric, harmonic, and Seiffert's means of two positive numbers 𝑎 and 𝑏 , respectively.