Region of variability for exponentially convex univalent functions

TL;DR

This paper characterizes the variability region of a specific complex expression for a class of univalent functions with exponential convexity, providing geometric insights and open problems in the field.

Contribution

It determines the region of variability for a complex functional of exponential convex univalent functions, extending understanding of their geometric properties.

Findings

Explicit description of the variability region $V(z_0, eta)$.

Geometric illustrations of the variability regions.

Proposed open problems for further research.

Abstract

For $\alpha\in\IC\setminus \{0\}$ let $\mathcal{E}(\alpha)$ denote the class of all univalent functions $f$ in the unit disk $\mathbb{D}$ and is given by $f(z)=z+a_2z^2+a_3z^3+\cdots$, satisfying $$ {\rm Re\,} \left (1+ \frac{zf''(z)}{f'(z)}+\alpha zf'(z)\right)>0 \quad {in ${\mathbb D}$}. $$ For any fixed $z_0$ in the unit disk $\mathbb{D}$ and $\lambda\in\overline{\mathbb{D}}$, we determine the region of variability $V(z_0,\lambda)$ for $\log f'(z_0)+\alpha f(z_0)$ when $f$ ranges over the class $$\mathcal{F}_{\alpha}(\lambda)=\left\{f\in\mathcal{E}(\alpha) \colon f''(0)=2\lambda-\alpha %\quad{and} f'''(0)=2[(1-|\lambda|^2)a+ %(\lambda-\alpha)^2 -\lambda\alpha] \right\}. $$ We geometrically illustrate the region of variability $V(z_0,\lambda)$ for several sets of parameters using Mathematica. In the final section of this article we propose some open problems.