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Multi-point variations of the Schwarz lemma with diameter and width conditions


Author: Dimitrios Betsakos
Journal: Proc. Amer. Math. Soc. 139 (2011), 4041-4052
MSC (2010): Primary 30C80
DOI: https://doi.org/10.1090/S0002-9939-2011-10954-7
Published electronically: March 28, 2011
MathSciNet review: 2823049
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Abstract: Suppose that $ f$ is holomorphic in the unit disk $ \mathbb{D}$ and $ f(\mathbb{D})\subset \mathbb{D}$, $ f(0)=0$. A classical inequality due to Littlewood generalizes the Schwarz lemma and asserts that for every $ w\in f(\mathbb{D})$, we have $ \vert w\vert\leq \prod_j \vert z_j(w)\vert$, where $ z_j(w)$ is the sequence of pre-images of $ w$. We prove a similar inequality by replacing the assumption $ f(\mathbb{D})\subset \mathbb{D}$ with the weaker assumption Diam $ f(\mathbb{D})=2$. This inequality generalizes a growth bound involving only one pre-image, proven recently by Burckel et al. We also prove growth bounds for holomorphic $ f$ mapping $ \mathbb{D}$ onto a region having fixed horizontal width. We give a complete characterization of the equality cases. The main tools in the proofs are the Green function and the Steiner symmetrization.


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Additional Information

Dimitrios Betsakos
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: betsakos@math.auth.gr

DOI: https://doi.org/10.1090/S0002-9939-2011-10954-7
Keywords: Holomorphic function, Schwarz lemma, Steiner symmetrization, capacity, Green function, inner function, diameter, width.
Received by editor(s): September 30, 2010
Published electronically: March 28, 2011
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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