Univalent functions with range restrictions (Q5938145)

Let \(\Sigma\) be the usual class of functions analytic and univalent in \(|z|> 1\) with series development about infinity of the form \(f(z) = z + a_0 + a_{-1}z^{-1} + \dots\). In this paper, the author investigates the extremal problem max Re \(a_{-1}\) in two special subclasses of \(\Sigma\). The first is the class \(\Sigma(\pm w_1)\) consisting of those \(f \in \Sigma\) which omit the values \(\pm w_1\). The second is the class \(\Sigma_b^0\) of all \(f \in \Sigma\) with \(a_0 = 0\) and such that the set of values omitted by \(f\) is of width greater than or equal to \(b\) in the imagionary direction. The paper includes a careful analysis of these problems and their relationship. A differential equation for the extremal function for the first problem is obtained in terms of \(w\) and \(\zeta = z + 1/z\). Interestingly,this is obtained by using the variational method for Bieberbach-Eilenberg functions. The quadratic differentials defined by this differential equation are analyzed most carefully. The work appears to be complete and correct. The problems are of some value and this paper deserves careful study.