Region of variability for functions with positive real part

Abstract

For $γ\in\IC$ such that $|γ|<π/2$ and $0\leqβ<1$, let ${\mathcal P}_{γ,β} $ 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 $λ\in\overline{\mathbb{D}}$, we shall determine the region of variability $V_{\mathcal{P}}(z_0,λ)$ for $\int_0^{z_0}P(ζ)\,dζ$ when $P$ ranges over the class $$ \mathcal{P}(λ) = \left\{ P\in{\mathcal P}_{γ,β} :\, P'(0)=2(1-β)λe^{-iγ}\cosγ\right\}. $$ 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.