## Uniqueness of a meromorphic function and its derivative.(English)Zbl 1056.30030

The authors consider sharing value problems of meromorphic functions and their derivatives. Let $$f(z)$$ be a meromorphic function. For a positive integer $$k$$ and $$a\in \mathbb{C} \cup\{\infty\}$$, $$N(r,a;f\mid\geq k)$$ denotes the counting function of $$a$$-points of $$f(z)$$ whose multiplicities are not less than $$k$$. They define $$N_p(r,a;,f) = \overline N(r,a;f) +\sum^p_{k=2}\overline N (r,a;f)\geq k)$$. For two meromorphic functions $$f(z)$$ and $$g(z)$$, they denote by $$E_k(a;f)$$ the set of all $$a$$-points whereby an $$a$$-point of multiplicity $$m$$ is counted $$m$$ times if $$m\leq k$$ and $$k + 1$$ times if $$m > k$$, where $$a\in \mathbb{C}\cup \{\infty\}$$ and $$k$$ is a nonnegative integer or infinity. If $$E_k(a;f) = E_k(a;g)$$, they say that $$f(z)$$ and $$g(z)$$ share the value $$a$$ with weight $$k$$. The authors used the symbol $$(a,k)$$ to express the situation of value sharing, namely, he wrote $$f(z)$$ and $$g(z)$$ share $$(a,k)$$ meaning that $$f(z)$$ and $$g(z)$$ share the value $$a$$ with weight $$k$$. It is remarked that $$f(z)$$ and $$g(z)$$ share value $$a$$ IM (ignoring multiplicity) or CM (counting multiplicity) if and only if $$f(z)$$ and $$g(z)$$ share $$(a,0)$$ or $$(a,\infty)$$, respectively. Further, they define $\delta_p(a,f)=1-\limsup_{r\to\infty}\;\frac{N_p(r,a;f)}{T(r,f)}\,,$ where $$p$$ is a positive integer.
The authors prove three theorems. I state the main theorem here. Let $$f(z)$$ be a nonconstant meromorphic function and $$k$$ be a positive integer. Let $$a(z)$$ $$(\not\equiv 0$$, $$\infty$$) be a meromorphic function such that $$T(r,a) = S(r,f)$$. If (i) $$a(z)$$ has no zero (pole) which is also a zero (pole) of $$f(z)$$ or $$f^{(k)}(z)$$ with the same multiplicity, (ii) $$f(z)-a(z)$$ and $$f^{(k)}(z)-a(z)$$ share $$(0,2)$$, (iii) $$2\delta_{2+k}(0,f)+(4+k)\Theta(\infty,f) > 5+k$$, then $$f(z)$$ coincides with $$f^{(k)}(z)$$. This result is an answer to the question posed by K.-W. Yu [JIPAM, J. Inequal. Pure Appl. Math. 4, No. 1, Paper No. 21, 7 p. (2003; Zbl 1021.30030)].

### MSC:

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

Zbl 1021.30030
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