×

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)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hayman, Meromorphic functions (1964)
[2] DOI: 10.1112/jlms/s2-20.3.457 · Zbl 0413.30025 · doi:10.1112/jlms/s2-20.3.457
[3] Gross, Complex Analysis (Proc. Conf. Univ. Kentucky, Lexington, KY, 1976) 599 pp 51– (1977)
[4] DOI: 10.1007/BF02835957 · Zbl 0799.30019 · doi:10.1007/BF02835957
[5] DOI: 10.2748/tmj/1178227619 · Zbl 0714.30028 · doi:10.2748/tmj/1178227619
[6] Yi, Complex Variables Theory Appl. 28 pp 13– (1995) · Zbl 0849.30022 · doi:10.1080/17476939508814834
[7] Yi, Complex Variables Theory Appl. 28 pp 1– (1995) · Zbl 0841.30027 · doi:10.1080/17476939508814833
[8] Yi, Science in China (Series A) 37 pp 802– (1994)
[9] Yi, Uniqueness theory of meromorphic functions (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.