Uniqueness of meromorphic functions with deficient poles. (English) Zbl 1070.30015

Let \(f\) be a nonconstant meromorphic function and let \(S\) be a set of distinct complex numbers. The set \(E_f(S)\) is the set of whole zeros of the equations \(f(z)-a=0, a\in S,\) counting multiplicities.
Let the equation \(z^n+az^{n-1}+b=0\) has no multiple roots and let \(B\) be the set of the roots of this equation. H. X. Yi (1996) proved that if \(f\) and \(g\) are nonconstant meromorphic functions and \(E_f(B) = E_g(B)\) then \(f=g\) or \[ f=\frac{ah(h^{n-1}-1)}{h^n-1}, \quad g=\frac{a(h^{n-1}-1)}{h^n-1}. \] The authors formulate additional conditions for \(f\) and \(g\) (two theorems) that bring \(f=g\) in the Yi’s theorem. They give examples demonstrated that the theorems are sharp. These theorems sharpen the theorem of M. Fang and W. Xu (1997).


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