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

摘要

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