On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function. (19th October 2011)

• Record Type: Journal Article
• Authors: Chu, Yu-Ming; Wang, Miao-Kun; Qiu, Ye-Fang
• Other Names: Aeyels Dirk Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2011(2011)
• Publication Date: 2011-10-19
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2011/697547
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : We prove that the double inequality( π / 2 ) (arth r / r ) 3 / 4 + α * r < 𝒦 ( r ) < ( π / 2 ) (arth r / r ) 3 / 4 + β * r holds for allr ∈ ( 0, 1 ) with the best possible constantsα * = 0 andβ * = 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.