A new type of sharp bounds for ratios of modified Bessel functions

TL;DR

This paper introduces novel, sharper bounds for ratios of modified Bessel functions of consecutive orders, derived through a qualitative Riccati equation analysis, with broad applicability in scientific computations.

Contribution

The paper presents a new method for deriving sharper bounds on Bessel function ratios using Riccati equation analysis, improving upon previous bounds.

Findings

New bounds are more accurate over a large parameter region.

The bounds outperform previous estimates in sharpness.

Method can be extended to other special functions.

Abstract

The bounds for the ratios of first and second kind modified Bessel functions of consecutive orders are important quantities appearing in a large number of scientific applications. We obtain new bounds which are accurate in a large region of parameters and which are shaper than previous bounds. The new bounds are obtained by a qualitative analysis of the Riccati equation satisfied by these ratios. A procedure is considered in which the bounds obtained from the analysis of the Riccati equation are used to define a new function satisfying a new Riccati equation which yields sharper bounds. Similar ideas can be applied to other functions.