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Note on Schwarz-Pick estimates for bounded and positive real part analytic functions. (English) Zbl 1133.30005
The authors prove the following extension of Schwarz’s lemma for the $n$th derivative: Let $f:D\to D$ be an analytic function on the unit disk $D$. Then $$ \vert f^{(n)}(z)\vert \leq \frac{n!(1-\vert f(z)\vert ^2)}{(1-\vert z\vert ^2)^n}(1+\vert z\vert )^{n-1},\;\;\;z\in D,\;\;\;n=1,2,3,\dots $$ The proof is partly based on the coefficient estimate $\vert a_n\vert \leq 1-\vert a_o\vert ^2$, where $f(z)=\sum_{n=0}^\infty a_nz^n$. This result improves other recent generalizations of Schwarz’s lemma. The authors prove also an analogous inequality for analytic functions with positive real parts and show that this inequality is asymptotically sharp.

30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
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