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On uniqueness of meromorphic functions with shared values in some angular domains. (English) Zbl 1045.30019

Two meromorphic functions \(f\) and \(g\) are said to share a value \(a\) if they have the same \(a-\)points. A classical result of Nevanlinna says that if \(f\) and \(g\) share five values, then \(f=g.\) Here it is shown that if \(f\) has finite lower order and if \(f\) or some derivative of \(f\) has a deficient value, then it suffices to assume that \(f\) and \(g\) share five values outside certain sectors in order to conclude that \(f=g.\) The size and configuration of these sectors depend on the lower order of \(f\) and on the deficiency. Such a result does not hold if \(f\) has infinite (lower) order, but for this case a similar result is given where \(f\) and \(g\) share five values outside certain rays. The proofs use, among other things, Nevanlinna theory in angular domains.

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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