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Abstract and Applied AnalysisVolume 2011 2011, Article ID 605259, 8 pages

Research Article

Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Department 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 𝑆𝑝𝑎,𝑏=𝑝𝑎−𝑏-arctan2𝑝𝑎−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 𝑆𝛼𝑎,𝑏<𝑇𝑎,𝑏<𝑆𝛽𝑎,𝑏 holds for all 𝑎,𝑏>0 with 𝑎≠𝑏, and give new bounds for the complete elliptic integrals of the second kind. Here, ∫𝑇𝑎,𝑏=2-𝜋0𝜋-2√𝑎2cos2𝜃+𝑏2sin2𝜃𝑑𝜃 denotes the Toader mean of two positive numbers 𝑎 and 𝑏.





Autor: Yu-Ming Chu, Miao-Kun Wang, Song-Liang Qiu, and Ye-Fang Qiu

Fuente: https://www.hindawi.com/



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