Tur\'an type inequalities for some Lommel functions of the first kind

TL;DR

This paper establishes Turán type inequalities for Lommel functions of the first kind using their infinite product representations, zero distribution properties, and connections to entire function classes.

Contribution

It introduces new Turán inequalities for Lommel functions of the first kind, leveraging their product representations and zero distribution analysis.

Findings

Derived Turán inequalities for Lommel functions of the first kind.

Connected Lommel functions to the Laguerre-Pólya class of entire functions.

Utilized zero distribution and sign properties to prove inequalities.

Abstract

In this paper certain Tur\'an type inequalities for some Lommel functions of the first kind are deduced. The key tools in our proofs are the infinite product representation for these Lommel functions of the first kind, a classical result of G. P\'olya on the zeros of some particular entire functions, and the connection of these Lommel functions with the so-called Laguerre-P\'olya class of entire functions. Moreover, it is shown that in some cases J. Steinig's results on the sign of Lommel functions of the first kind combined with the so-called monotone form of l'Hospital's rule can be used in the proof of the corresponding Tur\'an type inequalities.