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Unavoidable systems of functions. (English) Zbl 0873.30016

Let \(D\) denote a domain in the complex plane. A set of functions \(g_1(z),\dots,g_n(z)\) meromorphic in \(D\) is said to form an unavoidable system in \(D\), if whenever \(f\) is meromorphic (\(\not\equiv\) constant) in \(D\), at least one of the equations \(f(z)=g_i(z)\) \((i=1,2,\dots,n)\) has a root in \(D\). It was shown by L. A. Rubel and the reviewer [Mich. Math. J. 20, 289-296 (1974; Zbl 0263.30022)] that the minimum cardinality of an avoidable system in the complex plane is 3 and three distinct polynomials \(a_1\), \(a_2\), \(a_3\) such that \(a_1-a_2\) and \(a_2-a_3\) are not both constant form an unavoidable system.
In this paper, the authors show that the above result holds for an arbitrary plane domain or a Riemann surface. The authors illustrate how an unavoidable triplet can be constructed for an arbitrary non-compact Riemann surface. They also discuss the related problem about unavoidable sets of functions which avoid each other.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Citations:

Zbl 0263.30022
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References:

[1] DOI: 10.1007/BF02392595 · Zbl 0619.30027 · doi:10.1007/BF02392595
[2] Behnke, Entwicklungen analytischer Funktionen auf Riemannschen Fl?chen. Math. Ann 120 pp 430– (1949) · Zbl 0038.23502
[3] DOI: 10.2307/2048323 · Zbl 0741.32009 · doi:10.2307/2048323
[4] Rubel, Michigan Math J. 20 pp 289– (1973)
[5] Forster, Lectures on Riemann Surfaces (1981) · doi:10.1007/978-1-4612-5961-9
[6] DOI: 10.1007/BF01360812 · Zbl 0179.11402 · doi:10.1007/BF01360812
[7] Luecking, Complex analysis: a functional analysis approach (1984) · Zbl 0546.30002
[8] Gauthier, Canadian J. Math 28 pp 885– (1976) · Zbl 0321.46032 · doi:10.4153/CJM-1976-085-x
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[10] Hayman., Meromorphic functions (1964) · Zbl 0115.06203
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