Abstract

We first study conditions for a polynomial P(w) to satisfy the condition that P(f) = cP(g) implies f = g for any nonzero constant c and nonconstant meromorphic functions f and g on c. Next, we give some sufficient conditions for a finite set S to be a uniqueness range set, namely, to satisfy the condition that f-1(S) = g-1(S) implies f = g for any nonconstant meromorphic functions f and g on c. For a set S, we consider a polynomial P(w) of degree q := #S which vanishes on S. Let P'(w) have distinct k zeros d1,..., dk and assume that k ≥ 4. We show that, if q > 2k + 12, P(d) ≠ P(dm) (1 ≤ ℓ < mk) and P(d1) +...+ P(dk) ≠ 0, then S is a uniqueness range set, and discuss some other related subjects.

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