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.


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