Unicity theorems for meromorphic or entire functions III

THEOREM 1. Let S = {w | wn + aton~m + 6 = 0}, where n and m are two positive integers such that n and m have no common factors, m ^ 1 and n > 1m + 8, a and 6 are two nonzero constants such that the algebraic equation wn + awn~m + b = 0 has no multiple roots. Then S is a URS of meromorphic functions. https://doi.org/10.1017/S0004972700016737 Published online by Cambridge University Press

LEMMA 1. (See [5].) Let f be a nonconstant meromorphic function, and let P(f) be a polynomial in f of the form P(f) = aofn + ai/"" 1 + • • • + a n_i/ + an, wiere ao (^ 0), ai, ..., an are constants. Then T(r, P(f)) = nT(r, f) + S{r, f). In order to state the second lemma, we introduce the following notation. Let F be a meromorphic function. We denote by ni(r, 1/(JF — a)) the num- ber of simple a-points of F in \z\ ^ r. N\{r, 1/(F — a)) is defined in terms of ni(r, l/(F — a)) in the usual way (se

LEMMA 2. Let (F" 2F' \ (G" 2G' \ wiere F and G are two nonconstant meromorphic functions. If F and G share 1 CM, and H ^ 0, then PROOF: Suppose that ZQ is a simple 1-point of F. Let F{z) = 1 + Ol(* - *0) + oa(z - *o f +0({z- 20)3), G(x) = 1 + 61(2 - 20) + b2{z - 20)2 + O((* - 20)3), where aj ^ 0 and 61 ^ 0. Then an elementary calculation gives that H{z) = 0{z - 20), which proves that ZQ is a zero of H. Thus,

THEOREM 2 . Let S - {w | wn + aw""1 + 6 = 0}, where n > 10 is a pos- itive integer, a and b are two nonzero constants such that the algebraic equation wn + aw"'1 + b — 0 has no multiple roots. If f and g are two distinct nonconstant meromorphic functions satisfying Ef(S) — Eg(S), then 9 — y 1 hn-l y hn -1 ' wAere h is a nonconstant meromorphic function. PROOF: Let (33) F=-\r~\f