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The uniqueness theorems of meromorphic functions sharing three values and one pair of polynomials. (English) Zbl 1205.30029

A classical theorem by R. Nevanlinna states that if two meromorphic functions \(f\) and \(g\) share five values ignoring multiplicities, then \(f=g\).
The authors prove the following two theorems that improve similar results by R. Nevanlinna [Acta Math. 48, 367–391 (1926; JFM 52.0323.03)], G. G. Gundersen [J. Lond. Math. Soc., II. Ser. 20, 456–466 (1979; Zbl 0413.30025)], G. Brosch [Eindeutigkeitssätze für meromorphe Funktionen (RWTH Aachen, Math.-Naturwiss. Fak., Diss., Aachen) (1989; Zbl 0694.30027)], and others.
Theorem 1. Let \(f\) and \(g\) be two distinct non-constant meromorphic functions such that \(f\) and \(g\) share \( 0\), \(1\), and \(\infty\) counting multiplicities, and let \(P_1\) and \(P_2\) be two non-constant polynomials such that \(P_1\not\equiv P_2\). If \(f-P_1\) and \(g-P_2\) share 0 ignoring multiplicities, then \(f\) and \(g\) are transcendental meromorphic functions and satisfy one of the following three relations: (i) \(f+g=1\) with \(P_1+P_2=1\); (ii) \(f=\frac{P_1}{P_2} g\); (iii) \(f=\frac{P_1-1}{P_2-1}g +\frac{P_2-P_1}{P_2-1}\).
Theorem 2. Let \(f\) and \(g\) be two nonconstant entire functions such that \(f\) and \(g\) share \(0\) counting multiplicities, and let \(P_1\) and \(P_2\) be two nonconstant polynomials such that \(P_1\not\equiv P_2\). If \(f-P_1=0 \Rightarrow g-P_2= 0\), then \(f=g\).

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable (general theory)
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