Coefficients of univalent functions of the Gel'fer class
Let G denote the Gel'fer class of functions \(g(z)=1+a_ 1z+..\). analytic in the unit disk \(K_ 1\) with \(g(z_ 1)+g(z_ 2)\neq 0\) for all \(z_ 1,z_ 2\in K_ 1\). Let \(\tilde G\subset G\) denote the subclass of univalent functions. The author writes \(\tilde G(\)n) for the set of all functions \(g\in \tilde G\) such that \(| a_ n(g)| =\max_{\phi \in \tilde G}| a_ n(\phi)|\) \((n=1,2,...)\). Further, \(\tilde G_{loc}(n)\) denotes the set of all functions \(g\in \tilde G\) that yield a local maximum of \(| a_ n(g)|\), and \(\tilde G_{loc}(n,m)=\tilde G(n)\cap \tilde G_{loc}(m)\) (n\(\neq m)\). The set of all functions of the form \[ g_{\theta}(z)=(1+e^{i\theta}z)/(1-e^{i\theta}z)\quad (Im \theta =0) \] is denoted by P. The author proves that for each pair of natural numbers n and \(n+1\), exactly one of the following two assertions holds: \(\tilde G_{loc}(n,n+1)=P\) or \(\tilde G_{loc}(n,n+1)=\emptyset\). A corollary is that \(| a_ n(g)| \leq 2\) for each \(g\in \tilde G\) if \(\tilde G_{loc}(n,n+1)\neq \emptyset.\) This theorem is related to earlier work by the author. The author also finds a variational formula and a differntial equation satisfied by functions \(g\in \tilde G_{loc}(n)\).
scientific article; zbMATH DE number 3962087