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 features has a in basic terms conformal geometric formula when it comes to 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 because of M. Heins.

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5. 6. Let S and R be simply connected Riemann surfaces with analytic boundaries ∂S and ∂R, respectively; let Γ be an open and connected subset of ∂S ; and let R carry a complete regular conformal metric λ(w) |dw| with curvature bounded below and above by negative constants −cλ and −Cλ , respectively. Further, let f : S → R be an analytic map. Then the following conditions are equivalent. (i) f has an analytic extension across Γ such that f (Γ) ⊂ ∂R. (ii) For every ξ ∈ Γ, lim λ(f (z)) |f ′ (z)| |dz| = +∞ .

1, the function fˆ has an analytic extension across I . Thus f = ϕˆα ◦ fˆ has an analytic continuation across Γ and consequently to a £ whole neighborhood of ξ0 . 5. 6. Let S and R be simply connected Riemann surfaces with analytic boundaries ∂S and ∂R, respectively; let Γ be an open and connected subset of ∂S ; and let R carry a complete regular conformal metric λ(w) |dw| with curvature bounded below and above by negative constants −cλ and −Cλ , respectively. Further, let f : S → R be an analytic map.

13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, SpringerVerlag, Berlin–New York, 1997. [14] M. Heins, On a class of conformal metrics, Nagoya Math. J. 21 (1962), 1–60. [15] M. Heins, Some characterizations of finite Blaschke products of positive degree, J. Analyse Math. 46 (1986), 162–166. [16] M. Heins, A note concerning the lemma of Julia–Wolff–Carath´eodory, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 133–136. [17] P. , Cambridge University Press, Cambridge, 1998.

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