# Univalent Functions with Range Restrictions

### Siegfried Kirsch

Universität Halle-Wittenberg, Germany

## Abstract

Let $Σ$ be the class of functions $f(z)=z+a_{0}+a_{–1}z_{–1}+⋯$ analytic and univalent in $∣z∣>1$. In this paper we investigate the problem to maximize $Ra_{–1}$ in two subclasses of $Σ$: (i) the class of all functions $f∈Σ$ which omit two given values $±w_{1}(0<∣w_{1}∣<2)$ and (ii) the class of all functions $f∈Σ$ with $a_{0}=0$ which map onto regions of prescribed width $b_{f}=b(0<b<4)$ in the direction of the imaginary axis. We solve these problems by applying a variational method to a coefficient problem in two subclasses of univalent Bieberbach-Eilenberg functions which are equivalent to these problems.

## Cite this article

Siegfried Kirsch, Univalent Functions with Range Restrictions. Z. Anal. Anwend. 19 (2000), no. 4, pp. 1057–1073

DOI 10.4171/ZAA/998