A concavity property of the complete elliptic integral of the first kind

Abstract

We prove that the function Ga(x)=a−log⁡(1−x)K(x)(a∈R) is strictly concave on (0,1) if and only if a≥8/5. This solves a problem posed by Yang and Tian and complements their result that 1/Ga (a≥0) is strictly concave on (0,1) if and only if a=4/3. Moreover, we apply our concavity theorem to present several functional inequalities involving K. Among others, we prove that if a≥8/5, then 2aπ+1<a−log⁡(r′)K(r)+a−log⁡(r)K(r′)≤2a+log⁡(2)K(1/2) for all r∈(0,1), where r′=1−r2. Both bounds are sharp and the sign of equality holds if and only if r=1/2.

Topics & Concepts