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ESTIMATES OF THE SECOND DERIVATIVE OF BOUNDED ANALYTIC FUNCTIONS
Published online by Cambridge University Press: 03 June 2019
Abstract
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 $|z|$.
- 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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