Region of variability for exponentially convex univalent functions

Abstract

For $α\in\IC\setminus \{0\}$ let $\mathcal{E}(\alpha )$ denote the class of all univalent functions $f$ in the unit disk $\mathbb{D}$ and is given by $f(z)=z+a_2{z}^2+a_3{z}^3+\cdots$, satisfying $$ {\rm Re\,} \left (1+ \frac{zf''(z)}{f'(z)}+αzf'(z)\right)>0 \quad {in ${\mathbb D}$}. $$ 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 $\log f^{\prime }(z_0)+\alpha f(z_0)$ when $f$ ranges over the class $$\mathcal{F}_α(λ)=\left\{f\in\mathcal{E}(α) \colon f''(0)=2λ-α%\quad{and} f'''(0)=2[(1-|λ|^2)a+ %(λ-α)^2 -λα] \right\}. $$ 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.

11 pages and 8 figures