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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
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 $|w|\leq \prod _j |z_j(w)|$, 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
MR Author ID: 618946

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.