Results & Lemmas (7)
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THEOREM 1.
THEOREM 1. Let Sx = 1, w, w2,..., to""1, S2 = a, 6 and 5S = 0, wiiere a and b are constants such that ab ^ 0, an ^ bn, a2n ^ 1, 62n ^ 1 and…
THEOREM 1. Let Sx = {1, w, w2, ..., to""1}, S2 = {a, 6} and 5S = {0}, wiiere a and b are constants such that ab ^ 0, an ^ bn, a2n ^ 1, 62n ^ 1 and anbn / 1. Suppose that f and g are non-constant meromorphic functions satisfying Ef(Sj) = Ea{Sj) for j = 1, 2, 3. Then f = g.
THEOREM 2
THEOREM 2. Let Si and S2 be defined as in Theorem 1, and let S3 = oo. Suppose that f and g are non-constant meromorphic functions…
THEOREM 2 . Let Si and S2 be defined as in Theorem 1, and let S3 = {oo}. Suppose that f and g are non-constant meromorphic functions satisfying Ef(Sj) = Eg(Sj) for j = 1, 2, 3. Then f = g. From Theorem 2 we immediately obtain the following result, which answers the question posed by Goss.
THEOREM 3
THEOREM 3. Let Si and S2 be defined as in Theorem 1. Suppose that f and g are non-constant entire functions satisfying Ef(Sj) = Eg(Sj) for…
THEOREM 3 . Let Si and S2 be defined as in Theorem 1. Suppose that f and g are non-constant entire functions satisfying Ef(Sj) = Eg(Sj) for j = 1, 2. Then f = g. The following interesting result will be needed in the proof of our theorems. https://doi.org/10.1017/S0004972700016324 Published online by Cambridge University Press
THEOREM 4
THEOREM 4 • Let Si and S3 be defined as in Theorem 2. Suppose that f and g are non-constant meromorphic functions satisfying Ef(Sj) =…
THEOREM 4 • Let Si and S3 be defined as in Theorem 2. Suppose that f and g are non-constant meromorphic functions satisfying Ef(Sj) = Eg(Sj) for j = 1, 3. Then either f = eg, where cn = 1, or fg = d, where <f* = 1. 2. SOME LEMMAS
LEMMA 1
LEMMA 1. (See [8].) Let f and g be two non-constant meromorphic functions, and let ci, C2 and Cs be three non-zero constants. If c + C2</ =…
LEMMA 1 . (See [8].) Let f and g be two non-constant meromorphic functions, and let ci, C2 and Cs be three non-zero constants. If c\f + C2</ = C3, then , V) +N(r, f) + S(r, /). T(r, f) <
LEMMA 2
LEMMA 2. (See [6, 2].) Let /i, /2, • • •, fm be Hnearly independent meromorphic m /unctions satisfying ^ fj = 1 • Then for k = 1,2,...,m we…
LEMMA 2 . (See [6, 2].) Let /i, /2, • • •, fm be Hnearly independent meromorphic m /unctions satisfying ^ fj = 1 • Then for k = 1,2, ... ,m we have i r, y) + N(r, fk) + N(r, D)- where D denotes the Wronskain D = h fL / (TTI—l) /("**—l) / ( m — l ) 1 /2
LEMMA 3
LEMMA 3. (See [9].) Let /i,/2 and /j be three meromorphic functions satisfying J2 3 = 1 fj = 1, and let g = ~hlh, g-i. = I//2 and g3 =…
LEMMA 3 . (See [9].) Let /i,/2 and /j be three meromorphic functions satisfying J2 3 = 1 fj = 1, and let g\ = ~hlh, g-i. = I//2 and g3 = -/i//2- If /1, h and f3 are linearly independent, then and g$ are Hnearly independent. 3. PROOF OF THEOREM 4 By the assumption, from Nevanlinna's second fundamental theorem, we have n-l (1) (n - 1)T{T, 9)<YlN (r, -L-) + N(r, g) + S(r, g) fc=o (n + l)T(r,/)-$
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