Abstract and Applied Analysis
Volume 2011 (2011), Article ID 605259, 8 pages
http://dx.doi.org/10.1155/2011/605259
Research Article

Sharp Generalized Seiffert Mean Bounds for Toader Mean

Yu-Ming Chu,1 Miao-Kun Wang,1 Song-Liang Qiu,2 and Ye-Fang Qiu1

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

Received 4 June 2011; Revised 10 August 2011; Accepted 11 August 2011

Academic Editor: Detlev Buchholz

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 p∈[0,1], the generalized Seiffert mean of two positive numbers a and b is defined by Sp(a,b)=p(a-b)/arctan[2p(a-b)/(a+b)],  0<p≤1,  a≠b;  (a+b)/2,  p=0,  a≠b;  a,  a=b. In this paper, we find the greatest value α and least value β such that the double inequality Sα(a,b)<T(a,b)<Sβ(a,b) holds for all a,b>0 with a≠b, and give new bounds for the complete elliptic integrals of the second kind. Here, T(a,b)=(2/π)∫0π/2a2cos⁡2θ+b2sin⁡2θdθ denotes the Toader mean of two positive numbers a and b.