Function (biology)

Pure mathematics

Biology

Monotonic function

Economics

Logarithmically convex function

Evolutionary biology

Mathematics

Elliptic integral

Mathematical analysis

Subderivative

Geometry

Financial economics

Combinatorics

Convexity

Convex optimization

Concave function

Regular polygon

Convex function

Applicable Analysis and Discrete Mathematics

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 &lt; x &lt; 1) is the complete elliptic integral of the first kind.

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