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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Multi-point variations of the Schwarz lemma with diameter and width conditions
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by Dimitrios Betsakos PDF
Proc. Amer. Math. Soc. 139 (2011), 4041-4052 Request permission


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
  • Email:
  • Received by editor(s): September 30, 2010
  • Published electronically: March 28, 2011
  • Communicated by: Mario Bonk
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 4041-4052
  • MSC (2010): Primary 30C80
  • DOI:
  • MathSciNet review: 2823049