# zbMATH — the first resource for mathematics

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
##### Keywords:
radial limit; integral mean
Full Text: