On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function - Yu-Ming Chu; Miao-Kun Wang; Ye-Fang Qiu (Abstract and Applied Analysis)
We prove that the double inequality (π/2)(arthr/r)3/4+α*r<K(r)<(π/2)(arthr/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.