On a sum of modified Bessel functions

TL;DR

This paper investigates a sum of modified Bessel functions, establishing its mathematical properties, correcting a previous error, and proposing open problems for future research in the context of probability bounds.

Contribution

It provides new monotonicity, convexity, and inequality results for a specific Bessel function sum, and identifies an error in Kanter's earlier work.

Findings

Monotonicity and convexity properties established

Turán type inequalities derived

Correction of an error in Kanter's paper

Abstract

In this paper we consider a sum of modified Bessel functions of the first kind of which particular case is used in the study of Kanter's sharp modified Bessel function bound for concentrations of some sums of independent symmetric random vectors. We present some monotonicity and convexity properties for that sum of modified Bessel functions of the first kind, as well as some Tur\'an type inequalities, lower and upper bounds. Moreover, we point out an error in Kanter's paper [Ka] and at the end of the paper we pose an open problem, which may be of interest for further research.