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By Kraus D.

A boundary model of Ahlfors' Lemma is validated and used to teach that the classical Schwarz-Carathéodory mirrored image precept for holomorphic capabilities has a in basic terms conformal geometric formula by way of Riemannian metrics. This conformally invariant mirrored image precept generalizes clearly to analytic maps among Riemann surfaces and includes between different effects a characterization of finite Blaschke items as a result of M. Heins.

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Extra resources for A boundary version of Ahlfors` lemma, locally complete conformal metrics and conformally invariant reflection principles for analytic maps

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Notice that ∂R is a (not necessarily connected) real analytic manifold of (real) dimension 1. Now let f : → R be an analytic map. We say that f has an analytic extension across an open subarc Γ of ∂ with f (Γ) ⊆ ∂R, if there exists an analytic map F defined on a neighborhood U ⊆ of Γ, which maps into the Schottky double R = R ∪ ∂R ∪ R∗ such that F (Γ) ⊆ ∂R and F = f in ∩ U . Here R∗ denotes the mirror of R. 1. Let Γ be an open subarc of ∂ , let R be a simply connected Riemann surface with analytic boundary ∂R, let λ(w) |dw| be a complete regular conformal metric on R with curvature −cλ ≤ κλ ≤ −Cλ for some positive constants 250 D.

J. 25 (1958), 45–56. [7] C. Carath´eodory, Zum Schwarzschen Spiegelungsprinzip (Die Randwerte von meromorphen Funktionen), Comment. Math. Helv. 46 (1946), 263–278. 256 D. KRAUS, O. ROTH AND S. RUSCHEWEYH [8] H. Chen, On the Bloch constant, in Approximation, Complex Analysis, and Potential Theory, (ed. N. ), Kluwer Academic Publishers, Dordrecht, 2001, pp. 129–161. [9] R. Courant and D. Hilbert, Methoden der Mathematischen Physik II, Springer Berlin–Heidelberg– New York, 1968. [10] R. Fournier and St.

A) Note that λ(f (z)) |f ′ (z)| is the density of the pullback of the metric λ(w) |dw| under the map f , so f ∗ λ(z) |dz| = λ(f (z)) |f ′ (z)| |dz| is a conformal pseudometric on and therefore a well-defined function on . 1, it suffices to assume that λ(w) |dw| is a complete regular conformal metric with curvature bounded below. However, these assumptions cannot be weakened further. For instance, take R = , λ(w) |dw| = |dw| and any function holomorphic in a neighborhood of a point of the unit circle.

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