Univalency of certain transform of univalent functions

TL;DR

This paper investigates the univalency of a specific transform of univalent functions within the unit disc, focusing on conditions involving the function's coefficients and classes.

Contribution

It introduces a new transform of univalent functions and analyzes its univalency properties based on the original function's class and coefficients.

Findings

Conditions under which the transform is univalent are established.

The relationship between the original function's class and the transform's univalency is clarified.

Results extend understanding of function transforms in geometric function theory.

Abstract

We consider univalency problem in the unit disc $\mathbb D$ of the function $$g(z)=\frac{(z/f(z))-1}{-a_{2}},$$ where $f$ belongs to some classes of univalent functions in ${\mathbb D}$ and $a_{2}=\frac{f''(0)}{2}\neq 0$.