Uniqueness and value-sharing of meromorphic functions. (English) Zbl 0890.30019

Concerning the uniqueness and sharing values of meromorphic functions, many results about meromorphic functions that share more than or equal to two values have been obtained. In this paper, we study meromorphic functions that share only one value, and prove the following result: For \(n\geq 11\) and two meromorphic functions \(f(z)\) and \(g(z)\), if \(f^nf'\) and \(g^ng'\) share the same nonzero and finite value \(a\) with the same multiplicities, then \(f\equiv dg\) or \(g=c_1 e^{cz}\) and \(f=c_2e^{-cz}\), where \(d\) is an \((n+1)\)th root of unity, \(c\), \(c_1\) and \(c_2\) being constants. As applications, we solve some nonlinear differential equations.
Reviewer: X.Hua (Nanjing)


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