Redheffer type bounds for Bessel and modified Bessel functions of the first kind

TL;DR

This paper establishes new Redheffer type inequalities for Bessel and modified Bessel functions of the first kind, using monotonicity properties and power series techniques, and proposes a conjecture for future exploration.

Contribution

It introduces novel inequalities of Redheffer type for Bessel functions, expanding the theoretical understanding and providing tools for further research in special functions.

Findings

Derived new inequalities for Bessel functions

Proved a sharp inequality for the tangent function

Stated a conjecture for future investigation

Abstract

In this paper our aim is to show some new inequalities of Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We use also some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.