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Zbl 1133.30005
Dai, Shaoyu; Pan, Yifei
Note on Schwarz-Pick estimates for bounded and positive real part analytic functions.
(English)
[J] Proc. Am. Math. Soc. 136, No. 2, 635-640 (2008). ISSN 0002-9939; ISSN 1088-6826/e

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 $f(z)=\sum_{n=0}^\infty a_nz^n$. \par 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.
[Dimitrios Betsakos (Thessaloniki)]
MSC 2000:
*30C80 Maximum principle, etc. (one complex variable)

Keywords: Schwarz lemma; bounded analytic function; analytic function with positive real part

Cited in: Zbl 1243.32002 Zbl 1206.32001

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