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Sharp inequalities for the complete elliptic integral of the first kind

Let formula here be the complete elliptic integral of the first kind. Recently, S.-L. Qiu and M. K. Vamanamurthy have proved for all r∈(0, 1): formula here with r′=√1−r2. In this paper we establish that the converse inequality formula here holds 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.