On Alzer and Qiu's conjecture for complete elliptic integral and inverse hyperbolic tangent function (Q642727)

Summary: We prove that the double inequality \[ (\pi/2)(\operatorname{arth}r/r)^{3/4+\alpha^*r} <{\mathcal K}(r)< (\pi/2)(\operatorname{arth}r/r)^{3/4+\beta^*r} \] holds for all \(r\in(0, 1)\) with the best possible constants \(\alpha^* = 0\) and \(\beta^* = 1/4\), which answers to an open problem proposed by Alzer and Qiu. Here, \({\mathcal K}(r)\) is the complete elliptic integral of the first kind, and arth is the inverse hyperbolic tangent function.