Computational Methods and Function Theory 8 (2008), No. 2, 409--432
Copyright Heldermann Verlag 2008
A.F. Beardon
a.f.beardon@dpmms.cam.ac.uk , University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, England.
D. Minda
david.minda@math.uc.edu , University of Cincinnati, Department of Mathematical Sciences, Cincinnati, Ohio 45221-0025, U.S.A.
Schwarz's Lemma says that if a holomorphic self--map f of the unit disc fixes the origin then |f'(0)| ≤ 1. In 1931, Dieudonné extended this by showing that |f'(z)| ≤ 1 when |z|< √2 -1. We show that there is no analogue of Dieudonné's Lemma for a simply connected hyperbolic region unless the region is a disc and the fixed point is the centre. We also discuss the analogue of Dieudonné's result for multiply connected regions. In order to do this, we establish a local version of the Aumann-Carathéodory Rigidity Theorem, and we determine the maximal number of points fixed by some non-trivial conformal automorphism of a region in terms of its connectivity. More generally we investigate the topological relationship between the group of conformal automorphisms and its complement as subsets of the space of holomorphic self-maps of a hyperbolic region. For a simply connected hyperbolic region Ω we give sharp conformally invariant bounds on the hyperbolic distance between a holomorphic self-map f with a fixed point $a$ and the hyperbolic rotation about a through angle arg f'(a).
Keywords: Hyperbolic regions, Euclidean contractions, fixed points.
MSC 2000: Primary 30C99; Secondary 30F45, 47H09.
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