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.


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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