## On a question of Gross concerning uniqueness of entire functions.(English)Zbl 0905.30026

Let $$f$$ be an entire function, and let $$S\subset \mathbb{C}$$. We denote $$E_f(S) = f^{-1} (S)$$ (counting multiplicity) and $$\overline E_f (S)= f^{-1} (S)$$ (ignoring multiplicity). The author proves the theorem. Let the equation $$W^n (W-a)- b=0$$, $$n\geq 2$$, has no multiple zeros and let $$S_1= \{0\}$$, $$S_2=\{W \in\mathbb{C}: W^n(W-a)- b=0\}$$. Suppose that $$\overline E_f(S_1) = \overline E_g (S_1)$$, $$E_f(S_2) =E_g(S_2)$$ where $$f$$ and $$g$$ are entire functions. Then $$f \equiv g$$. Also he proves that the numbers $$\text{card} S_1=1$$, $$\text{card} S_2=3$$ are smallest for this question. Now we have full answer to the questions of Gross (1976). Earlier they gave a partial answer.

### MSC:

 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 30D20 Entire functions of one complex variable (general theory)

### Keywords:

entire function; questions of Gross; Nevanlinna theory
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### References:

  Hayman, Meromorphic functions (1964)  DOI: 10.1112/jlms/s2-20.3.457 · Zbl 0413.30025  Gross, Complex Analysis (Proc. Conf. Univ. Kentucky, Lexington, KY, 1976) 599 pp 51– (1977)  DOI: 10.1007/BF02835957 · Zbl 0799.30019  DOI: 10.2748/tmj/1178227619 · Zbl 0714.30028  Yi, Complex Variables Theory Appl. 28 pp 13– (1995) · Zbl 0849.30022  Yi, Complex Variables Theory Appl. 28 pp 1– (1995) · Zbl 0841.30027  Yi, Science in China (Series A) 37 pp 802– (1994)  Yi, Uniqueness theory of meromorphic functions (1995)
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