Unicity theorems for meromorphic or entire functions II

THEOREM 1. Let S = {w \ wn + awn~m +6 = 0}, wiere n and m are two positive integers such that n and m have no common factors and n ^ 2m + 5, a and b are two nonzero constants such that the algebric equation wn + awn~m + b = 0 ias no multiple roots. Then the set S is a URS of entire functions. By Theorem 1, we immediately obtain the following corollary.

COROLLARY 1. Let S — {w \ w7 +aw6 +b = 0}, wiere a and b are two non-zero constants such that b ^ —66(a/7) . Tien tie set S is a URS of entire functions with 7 elements. For meromorphic functions, we have the following result, which is an extension of

Theorem 1.

THEOREM 2 . Let S = {w \ wn+awn~m+b = 0}, wiere n and rn are two positive integers such that m ^ 2, n ^ 2m + 7 with n and m having no common factors, a and b are two nonzero constants such that the algebric equation wn + awn~m + b — 0 ias no multiple roots. Suppose that f and g are nonconstant meromorphic functions satisfying Ef(S) = Eg{S) and Ef({oo}) = Eg({oo}). Then f = g. By Theorem 2, we immediately obtain the following corollary.

COROLLARY 2. Let S — {w \ w11 -\-aw9 +b = 0}, wiere a and b are two non-zero constants such that b2 ^ —2299(a/ll) . Tien for any two nonconstant meromorphic https://doi.org/10.1017/S0004972700014635 Published online by Cambridge University Press

LEMMA 1. (see [6]). Let f and g be two nonconstant meromorpbic functions, and let c\ , c2 and C3 be three nonzero constants. If cif + c2g = c3, then T(r, f) < W(r, i ) + N(V, ±\ + N(r, f) + S(r, /).

LEMMA 2 . (see [4]). Let / 1 , f2, ..., fn be hnearly independent meromorphic functions satisfying n £/,=!• . 7 = 1 Tien for k — 1, 2, ..., n we have ) N(r> /*)+ N( r> D) - E N(r> fi) (T(r)) (r <£ E), where D denotes the WronsMan of the functions / 1 , / 2, ..., /„ and T(r) denotes the maximum of T(r, fj),j =

LEMMA 3 . (see [5]). Let f be a nonconstant meromorpbic function, and let P(f) be a polynomial in f of the form P(f) - 00/" + ai/"" 1 + • • • + an-i/ + an, where OQ (^ 0), OI, . . . , on are constants. Then T(r, P(f)) = nT(r, f) + 5(r, /). https://doi.org/10.1017/S0004972700014635 Published online by Cambridge University Press

LEMMA 4. fa, fa and fa are linearly dependent. PROOF: Suppose that fa, fa and fa are linearly independent. Applying Lemma 2 to the functions fj (j = 1, 2, 3), from (8) and (9) we have (10) ^jv(r, £) -N(r, ^ r, fa) ~ N(r, f3) S(r, /), where (11) From (5), (6) and (7) we have D = h h h /{ n n f" r-i n1 By looking at the zeros of / and g, from (5), (6), (7) and (11) we see that m)N (r, i ) - 2AT

THEOREM 3 . Let S = {w \ wn + aw*1'1 + 6 = 0}, wiere n > 8, and a and b are two nonzero constants such that the algebraic equation wn + awn~1 + 6 = 0 has no multiple roots. Suppose that f and g are two distinct nonconstant meromorphic functions satisfying Ef(S) = E9(S) and £/({oo}) = Ea({oo}). Then aHlH71-1 - 1) a ^ " " 1 - 1) f = H ^ l ""* 9 = 2 T - - 1 ' where H is a nonconstant meromorphic function. PROOF: Proceeding as in the proof of Theorem 2, we have