INEQUALITIES FOR THE RATIO OF COMPLETE ELLIPTIC INTEGRALS.

We present various inequalities for the complete elliptic integral of the first kind, K(r) = ∫π/20 1/√1-r2sin2(t)dt (0<r<1). Among others, we prove that the inequalities 1/1+1/4r < K(r)/K(√r) and K(√-r2)/K(√-r) < 2/1+√r are valid for all r ϵ (0,1). These estimates refine results published by Anderson, Vamanamurthy, and Vuorinen in 1990. [ABSTRACT FROM AUTHOR]