Variability regions of close-to-convex functions

TL;DR

This paper investigates the shape of the variability regions of certain functions related to close-to-convex functions, revealing that the full variability region of zf'(z)/f(z) is the complex plane minus zero, and introduces new forms for analysis.

Contribution

The paper proposes alternative forms for the variability regions of zf'(z)/f(z) and related functions, clarifying their shapes and properties.

Findings

Full variability region of zf'(z)/f(z) is the complex plane minus zero.

New forms of variability regions facilitate analysis of related logarithmic functions.

Improved understanding of the geometric behavior of close-to-convex functions.

Abstract

M. Biernacki gave concrete forms of the variability regions of $z/f(z)$ and $zf'(z)/f(z)$ of close-to-convex functions $f$ for a fixed $z$ with $|z|<1$ in 1936. The forms are, however, not necessarily convenient to determine the shape of the full variability region of $zf'(z)/f(z)$ over all close-to-convex functions $f$ and all points $z$ with $|z|<1.$ We will propose a couple of other forms of the variability regions and see that the full variability region of $zf'(z)/f(z)$ is indeed the complex plane minus the origin. We also apply them to study the variability regions of $\log[z/f(z)]$ and $\log[zf'(z)/f(z)].$