Entire functions that share one value with their derivatives (Q1906553)

Two meromorphic functions \(f\) and \(g\) share the complex value \(a\) if \(f(z) = a\) implies \(g(z) = a\) and vice versa. The value \(a\) is shared CM (counting multiplicities) if, in addition, \(f\) and \(g\) have the same multiplicities at each \(a\)-point. Let \(f\) be a nonconstant entire function. \textit{G. Jank}, \textit{E. Mues} and \textit{L. Volkmann} [Complex Variables, Theory Appl. 6, No. 1, 51-71 (1986; Zbl 0603.30037)] proved the following result: If \(f\) and \(f'\) share the value \(a \neq 0\) and if \(f(z) = a\) implies \(f''(z) = a\) then \(f \equiv f'\). In the paper under review it is shown that \(f''\) cannot be replaced by \(f^{(k)}\), \(k \geq 3\), in this theorem. the author proves the following generalisation (Theorem 1): If \(f\) and \(f'\) share the value \(a \neq 0\) CM and if \(f(z) = a\) implies \(f^{(n)} (z) = f^{(n + 1)} (z) = a\), \(n \geq 1\), then \(f \equiv f^{(n)}\). Two more results on entire functions sharing one value with their derivatives are presented.