On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function. (19th October 2011) - Chu, Yu-Ming; Wang, Miao-Kun; Qiu, Ye-Fang
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.