Univalent Functions with Range Restrictions

  • Siegfried Kirsch

    Universität Halle-Wittenberg, Germany


Let Σ\Sigma be the class of functions f(z)=z+a0+a1z1+f(z) = z + a_0 + a_{–1} z^{–1} + \cdots analytic and univalent in z>1|z| > 1. In this paper we investigate the problem to maximize Ra1\mathfrak R a_{–1} in two subclasses of Σ\Sigma: (i) the class of all functions fΣf \in \Sigma which omit two given values ±w1(0<w1<2)±w_1 (0 < |w_1| < 2) and (ii) the class of all functions fΣf \in \Sigma with a0=0a_0 = 0 which map onto regions of prescribed width bf=b(0<b<4)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