Abstract and Applied Analysis
Volume 2011 (2011), Article ID 697547, 7 pages
http://dx.doi.org/10.1155/2011/697547
Research Article

On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

Yu-Ming Chu, Miao-Kun Wang, and Ye-Fang Qiu

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

Received 23 May 2011; Accepted 16 August 2011

Academic Editor: Dirk Aeyels

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

We prove that the double inequality ( 𝜋 / 2 ) ( a r t h 𝑟 / 𝑟 ) 3 / 4 + 𝛼 ∗ 𝑟 < 𝒦 ( 𝑟 ) < ( 𝜋 / 2 ) ( a r t h 𝑟 / 𝑟 ) 3 / 4 + 𝛽 ∗ 𝑟 holds for all 𝑟 ∈ ( 0 , 1 ) with the best possible constants 𝛼 ∗ = 0 and 𝛽 ∗ = 1 / 4 , which answer to an open problem proposed by Alzer and Qiu. Here, 𝒦 ( 𝑟 ) is the complete elliptic integrals of the first kind, and arth is the inverse hyperbolic tangent function.