## On a question of Gross.(English)Zbl 1115.30029

The author mainly uses the notion of weighted sharing of sets to prove the following two interesting theorems:
Theorem 1: Let $$S_1=\left\{ z:z^n+az^{n-1}+b=0\right\}$$, $$S_2=\left\{0\right\}$$and $$S_3=\left\{\infty\right\}$$, where $$a$$ , $$b$$ are nonzero constants such that $$z^n+az^{n-1}+b=0$$ has no repeated root and $$n(\geq 4)$$ is an integer. If for two nonconstant meromorphic functions $$f$$ and $$g$$ $$E_f(S_1,4)=E_g(S_1,4), E_f(S_2,0)=E_g(S_2,0)$$ and $$E_f(S_3,\infty)=E_g(S_3,\infty)$$ and $$\Theta(\infty,f)+\Theta(\infty,g)>0$$ then $$f\equiv g$$.
Theorem 2: Let $$S_1=\left\{ z:z^n+az^{n-1}+b=0\right\}$$, $$S_2=\left\{0\right\}$$and $$S_3=\left\{\infty\right\}$$, where $$a$$ , $$b$$ are nonzero constants such that $$z^n+az^{n-1}+b=0$$ has no repeated root and $$n(\geq 3)$$ is an integer. If for two nonconstant meromorphic functions $$f$$ and $$g$$ $$E_f(S_1,6)=E_g(S_1,6), E_f(S_2,0)=E_g(S_2,0)$$ and $$E _f(S_3,\infty)=E_g(S_3,\infty)$$ and $$\Theta(\infty,f)+\Theta(\infty,g)>1$$ then $$f\equiv g$$.
The above mentioned uniqueness theorems improve the results of H. Qiu and M. L. Fang [J. Qufu Norm. Univ., Nat. Sci. 17, No. 3, 35–38 (1991; Zbl 0741.30027)] and several other results in this vein.

### MSC:

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

### Keywords:

meromorphic function; uniqueness; shared set; weighted sharing

Zbl 0741.30027
Full Text:

### References:

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