Modified Dini functions: monotonicity patterns and functional inequalities

TL;DR

This paper introduces new inequalities and properties of modified Dini functions, including monotonicity and convexity, by leveraging Bessel function representations and probability distributions.

Contribution

It presents novel functional inequalities and monotonicity results for modified Dini functions, expanding understanding of their mathematical properties.

Findings

Derived new Turán and Redheffer type inequalities.

Established complete monotonicity of a Dini function quotient.

Connected monotonicity to a new infinitely divisible probability distribution.

Abstract

We deduce some new functional inequalities, like Tur\'an type inequalities, Redheffer type inequalities, and a Mittag-Leffler expansion for a special combination of modified Bessel functions of the first kind, called modified Dini functions. Moreover, we show the complete monotonicity of a quotient of modified Dini functions by introducing a new continuous infinitely divisible probability distribution. The key tool in our proofs is a recently developed infinite product representation for a special combination of Bessel functions of the first, which was very useful in determining the radius of convexity of some normalized Bessel functions of the first kind.