Unicity theorems for meromorphic functions and their derivatives (Q558483)

The authors prove three theorems. Theorem 1. Let \(f\) be a nonconstant meromorphic function such that \(ff'\neq 0\), and let \(a(z)\not\equiv 0\) be a small function related to \(f\). If \(f(z) =a(z)\) whenever \(f^{'}(z) =a(z),\) then either \(f\equiv f^{'}\) or \(f\) assumes the form \[ f(z) =\frac{2a}{1-Ce^{-2z}}, \] where \(a, C\) are two non-zero constants and \(a(z) \equiv a.\) The authors show that the condition ``\(f(z) =a(z)\) whenever \(f^{'}(z) =a(z)\)'' cannot be replaced by ``\(f'(z) =a(z)\) whenever \( f(z) =a(z) .\)'' The case when \(f^{'}(z)\) is changed for \(f^{(k)}(z)\), \(k\geq 2,\) is investigated in other theorems. The theorems are generalization of certain previous results. These results are cited.