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 $\pm w_1(0<|w_1|<2)$ 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.