On Certain Bounds for Genus 0 Entire Functions

TL;DR

This paper derives an upper bound for genus 0 entire functions with only negative zeros using elementary methods, based solely on positive real axis data, with applications to Bessel and Airy functions.

Contribution

It introduces a simple, elementary approach to bounding genus 0 entire functions with negative zeros, expanding analytical tools for special functions.

Findings

Established an explicit upper bound on the right half-plane for such functions.

Applied the bound to modified Bessel functions and Airy functions.

Demonstrated the bound's effectiveness using only positive real axis information.

Abstract

Abstract. In this work we use an elementary method to derive an upper bound on the right half-plane for genus 0 entire functions if it has only negative zeros. The bound only uses information of the function on the positive real axis. Applications to the modified Bessel functions and the Airy function are also provided.