## Entire functions that share a polynomial with their derivatives.(English)Zbl 1096.30029

L. A. Rubel and C. C. Yang proved that if a nonconstant entire function $$f$$ and its derivative $$f'$$ share two distinct finite values, then $$f\equiv f'$$. Jank-Mues-Volkmann proved that if a nonconstant entire function $$f$$ and its derivative $$f'$$ share a finite value $$a$$ and $$f=a$$ implies that $$f''=a$$,then $$f\equiv f'$$. A natural question is to consider the case that $$f$$ and $$f'$$ share a small function $$a(z)$$ ($$T(r,a)=S(r,f)$$). In this paper, the author studied the entire functions sharing a polynomial with their derivatives. He proved that if an entire function $$f$$ and its derivative $$f'$$ share a nonconstant polynomial $$Q$$ with degree $$q<k$$ CM, and $$f(z)-Q(z)=0$$ implies that $$f^{(k)}(z)-Q(z)=0$$, then $$f\equiv f'$$. Remark: The statement of Lemma 2 in the paper is not right. It should be stated as “Let $$f$$ be an entire solution of the equation $a_n(z)f^{(n)}+a_{n-1}(z)f^{(n-1)}+\cdots+a_1(z)f'+a_0(z)f=0,(*)$ with polynomial coefficients $$a_0(z),\cdots,a_n(z)$$ such that $$a_0(z)\not\equiv 0$$ and $$a_n(z)\not\equiv 0$$, then $$f$$ is of finite order.” Please note that if $$a_n(z)$$ is not a constant, then the solution of the equation (*) may not be an entire function.

### MSC:

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

### Keywords:

Entire function; share value
Full Text:

### References:

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