Cross-product of Bessel functions: monotonicity patterns and functional inequalities

TL;DR

This paper investigates the properties of Dini functions and the cross-product of Bessel functions, focusing on monotonicity, inequalities, zero interlacing, and bounds for derivatives, using recent infinite product representations.

Contribution

It introduces new monotonicity patterns, inequalities, and zero interlacing results for Dini and Bessel-related functions, expanding understanding of their mathematical properties.

Findings

Established monotonicity patterns for cross-products of Bessel functions

Derived Redheffer-type inequalities for these functions

Proved interlacing properties of zeros and bounds for derivatives

Abstract

In this paper we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.