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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 $a\in \bbfC \cup\{\infty\}$, $N(r,a;f\mid\geq k)$ 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 $a\in \bbfC\cup \{\infty\}$ 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 {\it K.-W. Yu} [JIPAM, J. Inequal. Pure Appl. Math. 4, No. 1, Paper No. 21, 7 p. (2003; Zbl 1021.30030)].

MSC:
30D35Distribution of values (one complex variable); Nevanlinna theory
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