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.
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