Region of variability for functions with positive real part

Details Region of variability for functions with positive real part Region of variability for functions with positive real part

2010-06-04

arxiv.org/abs/1006.0906v1

Ann. Polon. Math. (2010)

Mathematics

Complex Variables

21 pages with 10 figures, to appear in 2010

Scientific paper

For $\gamma\in\IC$ such that $|\gamma|<\pi/2$ and $0\leq\beta<1$, let ${\mathcal P}_{\gamma,\beta} $ denote the class of all analytic functions $P$ in the unit disk $\mathbb{D}$ with $P(0)=1$ and $$ {\rm Re\,} \left (e^{i\gamma}P(z)\right)>\beta\cos\gamma \quad \mbox{ in ${\mathbb D}$}. $$ 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 $$ \mathcal{P}(\lambda) = \left\{ P\in{\mathcal P}_{\gamma,\beta} :\, P'(0)=2(1-\beta)\lambda e^{-i\gamma}\cos\gamma \right\}. $$ 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.

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