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Uniqueness of meromorphic functions that share one small function with their derivatives. (English) Zbl 1115.30034
G. G. Gundersen and L.-Z. Yang [J. Math. Anal. Appl. 223, No. 1, 88–95 (1998; Zbl 0911.30022)] proved the following theorem. Let $$f(z)$$ be a nonconstant entire function of finite order. If $$f(z)$$ and $$f'(z)$$ share one finite value CM, then $$(f'- a)/(f- a)= c$$ for some constant $$c$$. The authors consider some generalizations of the theorem above.
Namely, they consider whether there exist similar results for entire functions of infinite order, or meromorphic functions. Is it possible to exchange the constant $$a$$ into a small function $$a(z)$$ with respect to $$f(z)$$. They show the following theorem:
Let $$k\geq 1$$ be an integer. Let $$f(z)$$ be a nonconstant meromorphic function, and $$a(z)$$ be a nonzero small function with respect to $$f(z)$$. If $$f(z)- a(z)$$ and $$f^{(k)}(z)- a(z)$$ share the value 0 CM and $$f^{(k)}$$ and $$a(z)$$ do not have any common poles of same multiplicity and $2\delta(0, f)+ 4\Theta(\infty, f)> 5,\tag{1}$ then $$f(z)\equiv f^{(k)}(z)$$.
We note that the condition (1) is independent of $$k$$.
In connection with the authors’ result, we have, for instance, K.-W. Yu [On entire and meromorphic functions that share small functions with their derivatives, JIPAM, J. Inequal. Pure Appl. Math. 4, No. 1, Article 21, 7 pp. (2003; Zbl 1021.30030)].

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