Region of variability for functions with positive real part

Volume 99 / 2010

Saminathan Ponnusamy, Allu Vasudevarao Annales Polonici Mathematici 99 (2010), 225-245 MSC: Primary 30C45; Secondary 30C55,30C80. DOI: 10.4064/ap99-3-2

Abstract

For $\gamma\in\mathbb C$ 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 $$ {\rm Re}(e^{i\gamma}P(z))>\beta\cos\gamma\ \quad \hbox{in ${\mathbb D}$}. $$ 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.

Authors

  • Saminathan PonnusamyDepartment of Mathematics
    Indian Institute of Technology Madras
    Chennai 600 036, India
    e-mail
  • Allu VasudevaraoDepartment of Mathematics
    Indian Institute of Technology Madras
    Chennai-600 036, India
    e-mail

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