Sharp inequalities for the complete elliptic integral of the first kind

Letformula herebe the complete elliptic integral of the first kind. Recently, S.-L. Qiu and M. K. Vamanamurthy have proved for all r∈(0, 1):formula herewith r′=√1−r2. In this paper we establish that the converse inequalityformula hereholds for all r∈(0, 1). Moreover, we show that the constant factors ¼ and π/(4 log (2))−1 in (a 1) and (a 2) are best possible.