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 |