Univalent Functions with Range Restrictions

Abstract

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.