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On entire functions which share one value CM with their first derivative. (English) Zbl 0861.30032
This article is devoted to considering the following conjecture: Let $$f$$ be a nonconstant entire function of finite noninteger iterated order $\rho_1(f) = \lim\sup_{r\to \infty} \log \log T(r,f)/ \log r.$ If $$f$$, $$f'$$ share a value $$a\in \mathbb{C}$$ counting multiplicity, then for a constant $$c\in \mathbb{C} \backslash \{0\}$$, $$(f'-a)/(f-a)=c$$. Now, two theorems related to this conjecture will be proved: (1) The conjecture is true for $$a= 0$$. (2) If $$f$$ is nonconstant entire, $$N(r,{1 \over f'}) = S(r,f)$$ ad $$f$$, $$f'$$ share $$a=1$$ counting multiplicity, then for a constant $$c\in \mathbb{C} \backslash \{0\}$$, $$(f'-1)/(f-1) =c$$. Observe that the growth condition has been dropped in (2).
Reviewer: I.Laine (Joensuu)

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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##### References:
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