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Radial growth of the derivative of univalent functions. (English) Zbl 0549.30012
Let the function \(f(z)=z+a_ 2z^ 2+..\). be analytic univalent in \(| z| <1,\quad 0<r<1,\quad\alpha >0,\quad\lambda >0,\) and \(\gamma >\frac{1}{2}.\) the authors prove the following very interesting and useful results: \[ (1)\quad\int^{2\pi}_{0}|\log | f'(re^{i\theta})||^{\lambda}d\theta <A_{\lambda}(\log 1/(1- r))^{\lambda /2}, \]
\[ (2)\quad\lim \log_{r\to 1}| f'(re^{i\theta})| /(\log 1/(1-r))^{\gamma}=0 \] for almost all \(\theta\), and \[ (3)\quad\lim_{r\to 1}(1- r)^{\alpha}f'(re^{i\theta})=0 \] for almost all \(\theta\). (3) improves a long-standing result of W. Seidel and J. L. Walsh [Trans. Am. Math. Soc. 52, 128-216 (1942; Zbl 0060.220)] in the case \(\alpha ={1\over2}\).
Reviewer: D.Brannan

MSC:
30C55 General theory of univalent and multivalent functions of one complex variable
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