Region of variability for functions with positive real part

Abstract

For $γ\in\IC$ such that $|\gamma |<\pi /2$ and $0\le \beta <1$, let ${\mathcal{P}}_{\gamma ,\beta }$ denote the class of all analytic functions $P$ in the unit disk $\mathbb{D}$ with $P(0)=1$ and $$ {\rm Re\,} \left (e^{iγ}P(z)\right)>β\cosγ\quad \mbox{ in ${\mathbb D}$}. $$ For any fixed $z_0\in \mathbb{D}$ and $\lambda \in \overline{\mathbb{D}}$, we shall determine the region of variability $V_{\mathcal{P}}(z_0,\lambda )$ for ${\int }_0^{z_0}P(\zeta )d\zeta$ when $P$ ranges over the class $$\mathcal{P}(\lambda )=\left\{P\in {\mathcal{P}}_{\gamma ,\beta }:P^{\prime }(0)=2(1-\beta )\lambda e^{-i\gamma }\cos \gamma \right\}.$$ As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.

21 pages with 10 figures, to appear in 2010