Annales Academię Scientiarum Fennicę

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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