The radius of univalence of the reciprocal of a product of two analytic functions

TL;DR

This paper investigates the univalence radius of a reciprocal product of two analytic functions within the unit disk, providing sharp bounds and applications involving Bessel functions.

Contribution

It introduces new sharp bounds for the univalence radius of a specific reciprocal product of analytic functions, extending the understanding of univalent function subclasses.

Findings

Derived sharp univalence radius bounds for the reciprocal product

Established applications involving Bessel functions

Extended classical results to new subclasses of analytic functions

Abstract

Let ${\mathcal A}$ denote the family of all functions $f$ analytic in the open unit disk $\ID$ with the normalization $f(0)=0= f'(0)-1$ and ${\mathcal S}$ be the class of univalent functions from ${\mathcal A}$. In this paper, we consider radius of univalence of $F$ defined by $F(z)=z^{3}/(f(z)g(z))$, where $f$ and $g$ belong to some subclasses of ${\mathcal A}$ (for which $f(z)/z$ and $g(z)/z$ are non-vanishing in $\ID$) and, in some cases in precise form, belonging to some subclasses of ${\mathcal S}$. All the results are proved to be sharp. Applications of our investigation through Bessel functions are also presented.