## 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

sharing values

Zbl 0603.30037
Full Text:

### References:

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