Inequalities for the ratio of complete elliptic integrals

Abstract:

We present various inequalities for the complete elliptic integral of the first kind, \[ \mathcal {K}(r)=\int _0^{\pi /2} \frac {1}{\sqrt {1- r ^2\sin ^2(t)}}dt \quad {(0<r<1)}. \] Among others, we prove that the inequalities \[ \frac {1}{1+\frac {1}{4}r}<\frac {\mathcal {K}(r)}{\mathcal {K}(\sqrt {r})} \quad \mbox {and} \quad { \frac {\mathcal {K}(\sqrt {1-r^2})}{\mathcal {K}(\sqrt {1-r})}<\frac {2}{1+\sqrt {r}}} \] are valid for all $r\in (0,1)$. These estimates refine results published by Anderson, Vamanamurthy, and Vuorinen in 1990.