On a question of Gross. (English) Zbl 1115.30029

Banerjee, Abhijit

J. Math. Anal. Appl. 327, No. 2, 1273-1283 (2007).

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.

Reviewer: Subhas S. Bhoosnurmath (Dharwad)

MSC:

30D35

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

Keywords:

meromorphic function; uniqueness; shared set; weighted sharing

Citations:

Zbl 0741.30027

References:

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