On uniqueness of meromorphic functions sharing finite sets

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 d 1 ,..., d k and assume that k ≥ 4. We show that, if q > 2 k + 12, P ( d ℓ ) ≠ P ( d m ) (1 ≤ ℓ < m ≤ k ) and P ( d 1 ) +...+ P ( d k ) ≠ 0, then S is a uniqueness range set, and discuss some other related subjects.