Regions of variability for a class of analytic and locally univalent functions defined by subordination

TL;DR

This paper investigates the variability regions of the logarithmic derivative of functions within a specific class of analytic, locally univalent functions, extending known classes and precisely characterizing their behavior at fixed points.

Contribution

It determines the exact variability regions of '(z_0) for functions in the class al(A, B), generalizing the Janowski class of convex univalent functions.

Findings

Exact variability regions of '(z_0) are characterized.

The class al(A, B) properly contains the Janowski class.

Results extend known variability theorems for univalent functions.

Abstract

In this article we consider a family $\mathcal{C}(A, B)$ of analytic and locally univalent functions on the open unit disc $\ID=\{z :|z|<1\}$ in the complex plane that properly contains the well-known Janowski class of convex univalent functions. In this article, we determine the exact set of variability of $\log(f'(z_0))$ with fixed $z_0 \in \ID$ and $f"(0)$ whenever $f$ varies over the class $\mathcal{C}(A, B)$.