Annales Academić Scientiarum Fennicć
Mathematica
Volumen 33, 2008, 101-110

LOWER SCHWARZ-PICK ESTIMATES AND ANGULAR DEIRIVATIVES

J. Milne Anderson and Alexander Vasil'ev

University College London, Department of Mathematics
Gower Street, London WC1E 6BT, U.K.

University of Bergen, Department of Mathematics
Johannes Brunsgate 12, Bergen 5008, Norway; alexander.vasiliev 'at' math.uib.no

Abstract. The well-known Schwarz-Pick lemma states that any analytic mapping \phi of the unit disk U into itself satisfies the inequality |\phi'(z)| \leq (1-|\phi(z)|²) / (1-|z|²), z\in U. This estimate remains the same if we restrict ourselves to univalent mappings. The lower estimate is |\phi'(z)|\geq 0 generally or |\phi'(z)| > 0 for univalent functions. To make the lower estimate non-trivial we consider univalent functions and fix the angular limit and the angular derivative at some points of the unit circle. In order to obtain sharp estimates we make use of the reduced moduli of digons.

2000 Mathematics Subject Classification: Primary 30C35; Secondary 30C80.

Key words: Schwarz-Pick lemma, reduced modulus, digon, angular derivative.

Reference to this article: J.M. Anderson and A. Vasil'ev: Lower Schwarz-Pick estimates and angular derivatives. Ann. Acad. Sci. Fenn. Math. 33 (2008), 101-110.

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