Inequalities for modified Bessel functions and their integrals

TL;DR

This paper establishes new inequalities and bounds for integrals involving modified Bessel functions, including monotonicity results and a gamma function-based lower bound for $K_0(x)$, advancing mathematical understanding of these functions.

Contribution

It introduces novel inequalities and bounds for modified Bessel functions and their integrals, including a new lower bound for $K_0(x)$ involving gamma functions.

Findings

Established simple inequalities for integrals involving $I_{ u}(x)$ and $K_{ u}(x)$.

Proved a monotonicity property for $K_{ u}(x)$.

Derived a new lower bound for $K_0(x)$ involving gamma functions.

Abstract

Simple inequalities for some integrals involving the modified Bessel functions $I_{\nu}(x)$ and $K_{\nu}(x)$ are established. We also obtain a monotonicity result for $K_{\nu}(x)$ and a new lower bound, that involves gamma functions, for $K_0(x)$.