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Angular distribution of values of $f{f}^{\text{'}}$. (English) Zbl 0804.30024

The main result of the authors is the following: Theorem 1. Let $f\left(z\right)$ be an entire function of finite order $\lambda$, $argz={\theta }_{j}$ $\left(0\le {\theta }_{1}<{\theta }_{2}<\cdots <{\theta }_{q}<2\pi$, ${\theta }_{q+1}={\theta }_{1}+2\pi \right)$ be a finite number of rays, $n$ be the counting function of zeros of the function $f{f}^{\text{'}}-1$. If $f$ satisfies

$\underset{r\to \infty }{\overline{lim}}\frac{log{\sum }_{j=1}^{q}n\left(r,{\theta }_{j}+\epsilon ,{\theta }_{j+1}-\epsilon \right)}{logr}\le \rho$

for any small positive number $\epsilon$, then

$\lambda \le max\left(\frac{\pi }{{\theta }_{2}-{\theta }_{1}},\cdots ,\frac{\pi }{{\theta }_{q+1}-{\theta }_{q}},\rho \right)·$

Lemma 1 (there must be changed $O\left(1\right)$ to $O\left(logr\right)$ in the text of the lemma), which is cited by the authors as the result of Nevanlinna, is the hypothesis of Nevanlinna has yet to be demonstrated. I think that in place of the Lemma 1, the theorem 3.1 [A. A. Gol’dberg, J. V. Ostrovskii, The distribution of values of meromorphic functions. Moskva: Nauka (1970; Zbl 0217.10002) (Russian), chapter III, §3] is used.

##### MSC:
 30D15 Special classes of entire functions; growth estimates 30D35 Distribution of values (one complex variable); Nevanlinna theory