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 features has a only conformal geometric formula by way of Riemannian metrics. This conformally invariant mirrored image precept generalizes evidently to analytic maps among Riemann surfaces and comprises between different effects a characterization of finite Blaschke items because of M. Heins.

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

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To appear. [23] H. L. Royden, The Ahlfors–Schwarz Lemma: the case of equality, J. Analyse Math. 46 (1986), 261–270. ¨ [24] H. A. Schwarz, Uber einige Abbildungsaufgaben, J. Reine Angew. Math. 70 (1869), 105–120. [25] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993. [26] S. T. Yau, A general Schwarz Lemma for K¨ahler Manifolds, Amer. J. Math. 100 (1978), 197–203.

4 is just a very special case of Bland’s boundary Schwarz Lemma (which in its original form applies to higher dimensional situations). 1, λ(z) |dz| is a regular conformal pseudo-metric and the regularity of µ(z) |dz| is of no importance. A BOUNDARY VERSION OF AHLFORS’ LEMMA 249 By now, we have in hand enough technology to establish the following characterization of locally complete regular conformal metrics in terms of the boundary behavior of their density functions for smooth boundary sets. 6.

4. Let S = S0 ∪ ∂S be a bordered Riemann surface and let λ(z) |dz| be a pseudo-metric on S0 . Further, let {ϕα : Uα → } be the family of charts of S and let λα (u) |du| be the pseudo-metric λ(z) |dz| in the local parameter ϕα restricted to S0 . Then, if ξ ∈ ∂S belongs to Uα and lim λα (ϕα (z)) = +∞ , z→ξ then lim λα′ (ϕα′ (z)) = +∞ z→ξ for any other chart ϕα′ with ξ ∈ Uα′ . This enables us to say that a pseudo-metric tends to +∞ at a point ξ of the border ∂S , and we write λ(z) |dz| → +∞ as z → ξ in this case.

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