An extremal problem for harmonic self-mappings of the unit disk
It is known that \textit{P. Duren} and \textit{G. Schober} [Complex Variables, Theory Appl. 9, 153--168 (1987; Zbl 0606.30025)] developed a variational method for solving extremal problems over families of sense-preserving univalent harmonic functions which map the unit disc \(\mathbb{D}:=\{z:|z|<1\}\) onto given convex region. This method is most effective when specialised to harmonic functions which map \(\mathbb{D}\) onto \(\mathbb{D}\) (see also \textit{P. Duren} and \textit{G. Schober} [Proc. Am. Math. Soc. 106, No. 4, 967--973 (1989; Zbl 0693.30018)]. Let \(\Aut_H(\mathbb{D})\) be the class of harmonic automorphisms of the disc \(\mathbb{D}\). Assume that \(f\in\Aut_H (\mathbb{D})\) has a series expansion \(f(re^{i\varphi})= \sum^\infty_{n=-\infty}c_n r^{|n|}e^{in\varphi}\), \(r\in\langle 0,1)\). In this paper the author considers the extremal problem of finding the supremum of \(\text{Re}\,{\mathcal I}_n(f)\), where \(f\in\Aut_H(\mathbb{D})\) and \({\mathcal I}_n(f)=\sum^n_{k=1}(c_k+c_{-k})\). He proved that \(\text{Re} \,{\mathcal I}_n(f)<\frac {4}{\pi}{\mathcal W}_n(\frac{\pi}{n+1}), n=1,2,\dots\) where \({\mathcal W}_n(\alpha)=\sum^n_{k=1}\frac 1k\sin k\alpha\).
scientific article; zbMATH DE number 5129631