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Local variations and minimal area problem for Carathéodory functions. (English) Zbl 1057.30018

Let \(P\) denote the Carathéodory class of functions \(f\) analytic in the unit disc \(U=\{z:z<1\}\) and such that \(f(0)=1\), \(\text{Re}\,f(z)>0\) for \(z\in\mathbb{U}\). Let \(P_\alpha\), \(0\leq\alpha\leq 2\), be the subclass of functions \(f\in P\) such that \(| f'(0)|=\alpha\). The Dirichlet integral of \(fD(f)=\int_\mathbb{U}| f'|^2d\sigma\) measures the area of the image \(f(\mathbb{U})\) counting multiplicity of covering. We have \(D(f)=\pi \sum^\infty_{n=1} n| c_n(f)|^2 \geq\pi\alpha^2\) for \(f\in P_\alpha\), \(f(z)=1+c_1(f)z+\cdots\) and the function \(f_\alpha(z):= 1+ \alpha z\), \(z\in\mathbb{U}\), \(0\leq\alpha\leq 1\), belongs to \(P_\alpha\). Therefore \(f_\alpha\) minimizes \(D(f)\) in \(P_\alpha\) for \(0\leq\alpha \leq 1\). In this paper the authors obtain \(\min D(f)\), \(f\in P_\alpha\), for \(1< \alpha<2\) (Theorem 1.1). To solve this problem they develop a technique in the frame of classical complex analysis that explores symmetrization type geometric transformations and local boundary variations This method reduces the minimal area problem to a certain boundary value problem for analytic functions. In the paper there are also some comments on the known problem initiated by A. W. Goodman.

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
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