On bounded univalent functions whose ranges contain a fixed disk

Abstract:

Let $\mathcal {S}$ denote the standard normalized class of regular, univalent functions in $K = {K_1} = \{ z:|z| < 1\}$. Let $\mathcal {F}$ be a given compact subclass of $\mathcal {S}$. We consider the following two problems. Problem 1. Find $\max |{a_2}|$ for $f \in \mathcal {F}$. Problem 2. For $|z| = r < 1$, find the $\max \;(\min )|f(z)|$ for $f \in \mathcal {F}$. In this paper we are concerned with the subclass $\mathcal {S}_d^\ast (M) = \{ f \in \mathcal {S}:{K_d} \subset f(K) \subset {K_M}\}$. Through the use of the Julia variational formula and the Loewner theory we determine the extremal functions for Problems 1 and 2 for the class $\mathcal {S}_d^\ast (M)$, for all d, M such that $\tfrac {1}{4} \leqslant d \leqslant 1 \leqslant M \leqslant \infty$.