Region of variability for functions with positive real part

Saminathan Ponnusamy¹ ; Allu Vasudevarao²

¹Department of Mathematics Indian Institute of Technology Madras Chennai 600 036, India
²Department of Mathematics Indian Institute of Technology Madras Chennai-600 036, India

Annales Polonici Mathematici, Tome 99 (2010) no. 3, pp. 225-245

Résumé

For $\gamma\in\mathbb C$ such that $|\gamma|\pi/2$ and $0\leq\beta1$, let ${\mathcal P}_{\gamma,\beta} $ denote the class of all analytic functions $P$ in the unit disk $\mathbb{D}$ with $P(0)=1$ and $$ {\rm Re}(e^{i\gamma}P(z))>\beta\cos\gamma\ \quad \hbox{in ${\mathbb D}$}. $$ 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 $$ \mathcal{P}(\lambda) = \{ P\in{\mathcal P}_{\gamma,\beta} : P'(0)=2(1-\beta)\lambda e^{-i\gamma}\cos\gamma \}. $$ 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.

Détail

DOI : 10.4064/ap99-3-2

Keywords: gamma mathbb gamma leq beta mathcal gamma beta denote class analytic functions unit disk mathbb gamma beta cos gamma quad hbox mathbb fixed mathbb lambda overline mathbb shall determine region variability mathcal lambda int zeta zeta ranges class mathcal lambda mathcal gamma beta beta lambda i gamma cos gamma consequence present region variability subclasses univalent functions graphically illustrate region variability several sets parameters