## An entire function sharing a polynomial with its derivatives.(English)Zbl 1388.30036

The paper deals with unicity theorems for entire functions. Let $$f$$ be an entire function and $$a=a(z)$$ a polynomial of degree $$p$$. We let $$E(a;f)$$ denote the set of zeros of $$f-a$$ counted with multiplicities. We also denote by $$\overline{E}(a;f)$$ the set of distinct zeros of $$f-a$$. In the case where $$f$$ is a transcendental function, we set $$\text{deg}\,f=+\infty$$. We define the counting function $$n_A(r, f, a)$$ as the number of zeros of $$f$$ in $$A\cap \left\{|z|<r \right\}$$ with multiplicities, and let $N_A(r, f, a)=\int_0^r (n_A(t, f, a)-n_A(0, f, a))dt/t.$ Suppose that $$n \geq p+2$$ is a positive integer. Let $$A=\overline{E}(a;f)\setminus \overline{E}(a;f')$$ and $$B=\overline{E}(a;f')\setminus (\overline{E}(a;f^{(n)})\cap \overline{E}(a;f^{(n+1)}))$$. Then the authors prove the following unicity theorem: If $N_A(r, a, f)+N_B(r, a, f')=S(r, f)$ and each common zero of $$f-a$$ and $$f'-a$$ has the same multiplicity, then $$f=\lambda e^z$$ for some constant $$\lambda \neq 0$$.

### MSC:

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

### Keywords:

entire functions; uniqueness theorems
Full Text:

### References:

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