On Landau-Kolmogorov problem on the interval for absolutely monotonic functions

We solve the Landau-Kolmogorov problem and the Kolmogorov problem for three positive numbers on the class of functions which are absolutely monotonic on a finite interval.

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

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.

A characterization of absolutely monotonic (Δ) functions

Absolutely monotonic (?) function of order n are characterized in terms of n-dimensional totally increasing functions. Applications to n-copulas are presented.