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.


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


sharing values


Zbl 0603.30037
Full Text: DOI


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[2] G. JANK, E. MUES AND L. VOLKMAN, Meromorphe funktionen, die mit ihre ersten und Zweiten Ableitung einen endlichen Wert teilen, Complex Variables Theory Appl. (1), 6 (1986), 51-71. · Zbl 0603.30037
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