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Two remarks on bounded analytic functions. (English) Zbl 0581.30009

Let \(f(z)=\sum^{\infty}_{k=0}a_ kz^ k\) be analytic in \(D=\{z: | z| <1\}\) and \(\sup_{z\in D}| f(z)| \leq 1\). Let \(f^{(n)}\) be the \(n^{th}\) derivative of f. (i) Then \[ | f^{(n)}(z)| \leq n!(1-| f(z)|^ 2)/((1-| z|)^ n(1+| z|)),\quad z\in D. \] For every \(z\in D\) there exist functions \(f_ j\), \(j\in N\), bounded and analytic in D, such that \[ \lim_{j\to \infty}| f_ j^{(n)}(z)| /(1-| f_ j(z)|^ 2)=n!/(1-| z|)^ n(1+| z|). \] (ii) If in addition f satisfies \(| f(z)| <1\) and \(| f(z)+1| \geq s\), \(0<s<2\), then for every admissible choice of s the sharp inequality \[ \sum^{\infty}_{k=0}| a_ k|^ 2\leq 1-s^ 2Re(1-a_ 0)/(1+a_ 0) \] holds.
Reviewer: J.Waniurski

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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