Abstract
Let Ω be a simply connected domain contained in the right half plane, i.e. Ω ⊂ {w ∈ ℂ: Rew > 0}, and satisfying 1 ∈ Ω. Let P Ω be the conformal mapping of the unit disk \(\mathbb{D}\) onto Ω with P Ω(0) = 1 and P′Ω(0) > 0. Define \(\mathcal{C}V_\Omega\) to be the class of analytic functions f in \(\mathbb{D}\) such that
Then \(\mathcal{C}V_\Omega\) is a subclass of the normalized convex univalent functions in \(\mathbb{D}\). If Ω is starlike with respect to 1 and
in \(\mathbb{D}\), then we can determine the variability region \(\left\{ {f\left( {z_0 } \right):f \in \mathcal{C}\mathcal{V}_\Omega } \right\}\). As an application we shall show a subordination result and determine variability regions for uniformly convex functions.
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Dedicated to Professor Yoshihiro Mizuta on the occasion of his sixtieth birthday
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Yanagihara, H. Variability Regions for Families of Convex Functions. Comput. Methods Funct. Theory 10, 291–302 (2010). https://doi.org/10.1007/BF03321769
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DOI: https://doi.org/10.1007/BF03321769