Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind

By Miao-Kun Wang, Hong-Hu Chu, Yong-Min Li, Y. Chu. Applicable Analysis and Discrete Mathematics, 2020

67 citations recorded by the source corpus.

In the article, we prove that the function x ? (1-x)pK(?x) is logarithmically concave on (0,1) if and only if p ? 7/32, the function x ? K(?x)/log(1+4/?1-x) is convex on (0,1) and the function x ? d2/dx2 [K(?x)- log (1+4/?1-x) is absolutely monotonic on (0,1), where K(x) = ??/20 (1-x2 sin2t)-1/2 dt (0 < x < 1) is the complete elliptic integral of the first kind.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1407a7a56a0a56bc1457948464a79c7f90611407