On Tur\'an type inequalities for modified Bessel functions

TL;DR

This paper explores inequalities related to modified Bessel functions, showing their equivalence to Turán type inequalities, introduces new inequalities, and discusses their implications for stability analysis in certain differential equations.

Contribution

It establishes the equivalence of recent inequalities to Turán type inequalities and presents new inequalities with applications in stability studies.

Findings

Certain inequalities are equivalent to Turán inequalities

Product of Bessel functions' inequalities is decreasing with order

New Turán type inequalities are proposed

Abstract

In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Tur\'an type inequalities for these functions. Moreover, we present some new Tur\'an type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has application in the study of stability of radially symmetric solutions in a generalized FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note a conjecture is posed, which may be of interest for further research.