Abstract
By improving a generalization of Borel's theorem, the authors have been able to show that there exists a finite set S with 15 elements such that for any two nonconstant meromorphic functions f and g the condition Ef(S)=Eg(S) implies f≡g. As a special case this also answers an open question posed by Gross [1] about entire functions, and has improved some results obtained recently by Yi [10]. In the last section, the uniqueness polynomials of meromorphic functions which is related to the unique range sets has been studied. A necessary and sufficient condition for a polynomial of degree 4 to be a uniqueness polynomial is obtained.
