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A Schwarz lemma for the modulus of a vector-valued analytic function


Author: Miroslav Pavlović
Journal: Proc. Amer. Math. Soc. 139 (2011), 969-973
MSC (2010): Primary 30C80, 46E15
DOI: https://doi.org/10.1090/S0002-9939-2010-10578-6
Published electronically: October 6, 2010
MathSciNet review: 2745648
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Abstract: It is proved that

$\displaystyle \vert\nabla\vert f\vert(z)\vert\le \frac{1-\vert f(z)\vert^2}{1-\vert z\vert^2}, \quad z\in \mathbb{D}, $

where $ f:\mathbb{D}\mapsto \mathbb{B}_k$ is an analytic function from the unit disk $ \mathbb{D}$ into the unit ball $ \mathbb{B}_k\subset \mathbb{C}^k.$ Applications to the Lipschitz condition of the modulus of a $ \mathbb{C}^k$-valued function are given.


References [Enhancements On Off] (What's this?)

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Additional Information

Miroslav Pavlović
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11001 Belgrade, p.p. 550, Serbia
Email: pavlovic@matf.bg.ac.yu

DOI: https://doi.org/10.1090/S0002-9939-2010-10578-6
Keywords: Schwarz lemma, Lipschitz condition, vector-valued analytic functions
Received by editor(s): March 22, 2010
Published electronically: October 6, 2010
Additional Notes: Supported by Ministarstvo nauke Srbije, Project ON144010
Communicated by: Richard Rochberg
Article copyright: © Copyright 2010 American Mathematical Society

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