A boundary version of Ahlfors` lemma, locally complete by Kraus D.

By Kraus D.

A boundary model of Ahlfors' Lemma is proven and used to teach that the classical Schwarz-Carathéodory mirrored image precept for holomorphic services has a in basic terms conformal geometric formula when it comes to Riemannian metrics. This conformally invariant mirrored image precept generalizes evidently 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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Proc. Cambridge Phil. Soc. 130 (2001), 353–364. [12] O. Frostman, Sur les produits de Blaschke, Fysiogr. S¨allsk. Lund F¨orh. 12 (1942), 169–182. [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.

1) holds for every ξ ∈ Γ. 1. 4 (Bland’s boundary Schwarz Lemma). Let Ω ⊆ be a domain and let Γ be a smooth subset of ∂Ω. Further, let λ(z) |dz| be a regular conformal metric on Ω with κλ ≥ −cλ , and let µ(z) |dz| be a regular conformal pseudo-metric on Ω with κµ ≤ −Cµ for some positive constants cλ and Cµ . If λ(z) |dz| is locally complete near Γ, then Cµ λ(z) lim inf ≥ z→ξ µ(z) cλ for every ξ ∈ Γ. 5. 4 is just a very special case of Bland’s boundary Schwarz Lemma (which in its original form applies to higher dimensional situations).

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