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Uniqueness of non-linear differential polynomials sharing 1-points. (English) Zbl 1073.30022
Given a non-constant meromorphic function \(f\), let \(E_{k)}(a,f)\) be the set of all \(a\)-points of \(f\) of multiplicity \(\leq k\), each point counted by its multiplicity. In [Indian J. Pure Appl. Math. 32, No. 9, 1343–1348 (2001; Zbl 1005.30023)], Fang and Hong proved that whenever \(f\), \(g\) are transcendental entire, \(n\geq11\) is an integer and \(f^n(f-1)f'\) and \(g^n(g-1)g'\) share the value one CM, then \(f=g\). In the present paper, this result is improved by showing that \(n\geq11\) can be replaced by \(n\geq 7\) and CM-sharing by \(E_{3)}(1,f^n(f-1)f')=E_{3)}(1,g^n(g-1)g')\). If \(f\), \(g\) are meromorphic, then the conclusion holds (a) for \(n\geq12\) in general, and (b) for \(n\geq11\), whenever \(\theta(\infty,f)>0\), \(\theta(\infty,g)>0\) and \(\theta(\infty,f)+\theta(\infty,g)>\frac4{n+1}\). The proofs need a careful analysis of the counting functions.

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