Finiteness of entire functions sharing a finite set

For a finite set S = {a(1),..., a(q)}, consider the polynomial P-S(w) = (w - a(1)) (w - a(2)) ... (w - a(q)) and assume that P-S' (w) has distinct k zeros. Suppose that Ps(w) is a uniqueness polynomial for entire functions, namely that, for any nonconstant entire functions phi and psi, the equality P-S(phi) = cP(S)(psi) implies phi = psi, where c is a nonzero constant which possibly. depends on phi and psi. Then, under the condition q > k + 2, we prove that, for any given nonconstant entire function 9, there exist at most (2q-2)/(q - k - 2) nonconstant entire functions f with f*(S) = g*(S), where f*(S) denotes the pull-back of S considered as a divisor. Moreover, we give some sufficient conditions of uniqueness polynomials for entire functions.