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Value regions of univalent self-maps with two boundary fixed points. (English) Zbl 1393.30021
Let \(f\) be a holomorphic self-map of the unit disk \(\mathbb D\). A point \(\sigma \in \partial \mathbb D\) is called a boundary regular fixed point if \(f(\sigma) = \sigma\) and the angular derivative \(f'(\sigma)\) is finite. It is known that if \(f \in \mathrm{Hol}(\mathbb D, \mathbb D)\) has no fixed points in \(\mathbb D\), then it has a unique boundary regular fixed point \(\tau\) with \(f'(\tau) \leq 1\). Such a point \(\tau\) is called (boundary) Denjoy-Wolff point. The authors find the exact value region \({\mathcal V}(z_0, T)\) of the point evaluation functional \(f \to f(z_0)\) over the class of all univalent self-maps \(f\) of \(\mathbb D\) having a boundary regular fixed point at \(\sigma = -1\) with \(f'(-1) = e^{T}\) and the Denjoy-Wolff point at \(\tau = 1\).

MSC:
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30C35 General theory of conformal mappings
30C55 General theory of univalent and multivalent functions of one complex variable
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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