ESTIMATES OF THE SECOND DERIVATIVE OF BOUNDED ANALYTIC FUNCTIONS
2019 ◽
Vol 100
(3)
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pp. 458-469
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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 $|z|$.
2016 ◽
Vol 103
(1)
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pp. 104-115
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2001 ◽
Vol 25
(12)
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pp. 771-775
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2000 ◽
Vol 24
(9)
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pp. 577-581
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