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Angular distribution of values of $ff'$. (English) Zbl 0804.30024
The main result of the authors is the following: Theorem 1. Let $f(z)$ be an entire function of finite order $\lambda$, $\arg z = \theta\sb j$ $(0 \le \theta\sb 1 < \theta\sb 2 < \cdots < \theta\sb q < 2 \pi$, $\theta\sb{q + 1} = \theta\sb 1 + 2 \pi)$ be a finite number of rays, $n$ be the counting function of zeros of the function $ff'-1$. If $f$ satisfies $$\varlimsup\sb{r \to \infty} {\log \sum\sp q\sb{j = 1} n(r, \theta\sb j + \varepsilon, \theta\sb{j + 1} - \varepsilon) \over \log r} \le \rho$$ for any small positive number $\varepsilon$, then $$\lambda \le \max \left( {\pi \over \theta\sb 2 - \theta\sb 1}, \dots, {\pi \over \theta\sb{q + 1} - \theta\sb q}, \rho \right).$$ Lemma 1 (there must be changed $O(1)$ to $O (\log r)$ 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 [{\it A. A. Gol’dberg}, {\it J. V. Ostrovskii}, The distribution of values of meromorphic functions. Moskva: Nauka (1970; Zbl 0217.10002) (Russian), chapter III, § 3] is used.

30D15Special classes of entire functions; growth estimates
30D35Distribution of values (one complex variable); Nevanlinna theory