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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 19, Issue 4, 2000, pp. 1057–1073**

**DOI: 10.4171/ZAA/998**

Published online: 2000-12-31

Univalent Functions with Range Restrictions

Siegfried Kirsch^{[1]}(1) Universität Halle-Wittenberg, Germany

Let $\Sigma$ be the class of functions $f(z) = z + a_0 + a_{–1} z^{–1} + \cdots$ analytic and univalent in $|z| > 1$. In this paper we investigate the problem to maximize $\mathfrak R a_{–1}$ in two subclasses of $\Sigma$: (i) the class of all functions $f \in \Sigma$ which omit two given values $±w_1 (0 < |w_1| < 2)$ and (ii) the class of all functions $f \in \Sigma$ 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.

*Keywords: *Conformal mappings, coefficient problems, width of domains, Bieberbach-Eilenberg functions

Kirsch Siegfried: Univalent Functions with Range Restrictions. *Z. Anal. Anwend.* 19 (2000), 1057-1073. doi: 10.4171/ZAA/998