Region of Variability for Exponentially Convex Univalent Functions

Abstract

For ${\alpha\in\mathbb C{\setminus}\{0\}}$ 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

$${\rm Re}\left (1+ \frac{zf''(z)}{f'(z)}+\alpha zf'(z)\right ) > 0 \quad {\rm in }\,{\mathbb D}.$$

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, λ) for log f′(z ₀) + αf(z ₀) when f ranges over the class

$$\mathcal{F}_{\alpha}(\lambda)=\left\{f\in\mathcal{E}(\alpha) \colon f''(0)=2\lambda-\alpha\right\}.$$

We geometrically illustrate the region of variability V(z ₀, λ) for several sets of parameters using Mathematica. In the final section of this article we propose some open problems.