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Region of Variability for Some Subclasses of Univalent Functions

  • A. VasudevaraoEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 91)

Abstract

Let \(\fancyscript{A}\) denote the class of analytic functions \(f\) in the unit disk \(\mathbb {D}\) with \(f(0)=0\) and \(f'(0)=1\). Let \(\fancyscript{S}\) denote the class of univalent functions in \(\fancyscript{A}\). Let \(\widetilde{\fancyscript{F}}\) (for example, class of starlike, convex, close-to-convex, spiralike, etc.) be any arbitrary subfamily of \(\fancyscript{S}\) and \(z_0\in \mathbb {D}\) then upper and lower estimates of \(|f(z_0)|\), \(|f'(z_0)|\) and \(\mathrm{Arg} f'(z_0)\) for all \(f\in \widetilde{\fancyscript{F}}\) are respectively called a growth theorem, a distortion theorem and a rotation theorem at \(z_0\) for \(\widetilde{\fancyscript{F}}\). These estimates deal only with absolute values of \(f(z_0)\) and \(f'(z_0)\) or with the argument of \(f'(z_0)\). The aim of this paper is to give a survey on regions of variability of \(f(z_0)\) or \(f'(z_0)\) or \(\log f'(z_0)\) when \(f\) ranges over some well-known subclasses of \(\fancyscript{S}\). As a consequence, we present the sharp Pre-Schwarzian norm and Block semi-norm for some of the subclasses of \(\fancyscript{S}\). We also graphically illustrate the region of variability for several sets of parameters.

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Copyright information

© Springer India 2014

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

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