Functional inequalities for modified Bessel functions

TL;DR

This paper establishes mean value inequalities for modified Bessel functions using bounds on their logarithmic derivatives, and applies these results to prove the log-concavity of the gamma-gamma distribution's CDF.

Contribution

It introduces new mean value inequalities for modified Bessel functions and links these to Turán type inequalities, with applications to distribution properties.

Findings

Proved mean value inequalities for Bessel functions

Established log-concavity of gamma-gamma distribution CDF

Connected inequalities to Turán type bounds

Abstract

In this paper our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kinds. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Tur\'an type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind we prove that the cumulative distribution function of the gamma-gamma distribution is log-concave. At the end of this paper several open problems are posed, which may be of interest for further research.