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Entire functions that share one value with their derivatives. (English) Zbl 0840.30013
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. G. Jank, E. Mues and 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.

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