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ESTIMATES OF THE SECOND DERIVATIVE OF BOUNDED ANALYTIC FUNCTIONS

Part of: Riemann surfaces Geometric function theory

Published online by Cambridge University Press:  03 June 2019

GANGQIANG CHEN Open the ORCID record for GANGQIANG CHEN [Opens in a new window]
GANGQIANG CHEN*
Affiliation:
Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan email cgqmath@ims.is.tohoku.ac.jp
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Abstract

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Assume a point $z$ lies in the open unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and $f$ is an analytic self-map of $\mathbb{D}$ fixing 0. Then Schwarz’s lemma gives $|f(z)|\leq |z|$, and Dieudonné’s lemma asserts that $|f^{\prime }(z)|\leq \min \{1,(1+|z|^{2})/(4|z|(1-|z|^{2}))\}$. We prove a sharp upper bound for $|f^{\prime \prime }(z)|$ depending only on $|z|$.

Keywords

bounded analytic functionsSchwarz’s lemmaDieudonné’s lemma

MSC classification

Primary: 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination
Secondary: 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 458 - 469
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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