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Uniqueness theorems of meromorphic functions of a certain form. (English) Zbl 1186.30037

Author’s abstract: We show that, for any entire function \(f\) and all positive integers \(m, n\in \mathbb{N}\), possibly except for the special case \(m=n=1\), the function \(f^m(f^n-1)f^{\prime}\) has no non-zero finite Picard value. Furthermore, we show that, for any two non-constant meromorphic functions \(f\) and \(g\), if \(f^m(f^n-1)f^{\prime}\) and \(g^m(g^n-1)g^{\prime}\) share the value 1 weakly, then \(f\equiv g\) provided that \(m\) and \(n\) satisfy some conditions. In particular, if \(f\) and \(g\) are entire, then the restrictions on \(m\) and \(n\) can be greatly reduced.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D20 Entire functions of one complex variable (general theory)
30D30 Meromorphic functions of one complex variable (general theory)
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