Variability Regions for Families of Convex Functions

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 $$1 + \frac{{zf''(z)}} {{f'(z)}} \in \Omega for all z \in \mathbb{D} and f(0) = f'(0) - 1 = 0.$$ 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 $$Re\left( {P\Omega (z) - 1 + \frac{{zP'_\Omega (z)}} {{P_\Omega (z) - 1}}} \right) > 0$$ 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.