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Results & Lemmas (7)
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Theorem 1
Theorem 1. Let, and, where a and b are constants such that,,, and. Suppose that f and g are non-constant meromorphic functions satisfying…
Theorem 1. Let $S_1 = \{1, w, w^2, ..., w^{n-1}\}$ , $S_2 = \{a, b\}$ and $S_3 = \{0\}$ , where a and b are constants such that $ab \neq 0$ , $a^n \neq b^n$ , $a^{2n} \neq 1$ , $b^{2n} \neq 1$ and $a^nb^n \neq 1$ . Suppose that f and g are non-constant meromorphic functions satisfying $E_f(S_j) = E_g(S_j)$ for j = 1, 2, 3. Then f = g.
Theorem 2
Theorem 2. Let and be defined as in Theorem 1, and let. Suppose that f and g are non-constant meromorphic functions satisfying for j = 1,…
Theorem 2. Let $S_1$ and $S_2$ be defined as in Theorem 1, and let $S_3 = \{\infty\}$ . Suppose that f and g are non-constant meromorphic functions satisfying $E_f(S_j) = E_g(S_j)$ 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 and be defined as in Theorem 1. Suppose that f and g are non-constant entire functions satisfying for j = 1, 2. Then f = g.…
Theorem 3. Let $S_1$ and $S_2$ be defined as in Theorem 1. Suppose that f and g are non-constant entire functions satisfying $E_f(S_j) = E_g(S_j)$ for j = 1, 2. Then f = g.
The following interesting result will be needed in the proof of our theorems.
Theorem 4
Theorem 4. Let and be defined as in Theorem 2. Suppose that f and g are non-constant meromorphic functions satisfying for j = 1, 3. Then…
Theorem 4. Let $S_1$ and $S_3$ be defined as in Theorem 2. Suppose that f and g are non-constant meromorphic functions satisfying $E_f(S_j) = E_g(S_j)$ for j = 1, 3. Then either f = cg, where $c^n = 1$ , or fg = d, where $d^n = 1$ .
Lemma 1
Lemma 1. (See [8].) Let f and g be two non-constant meromorphic functions, and let, and be three non-zero constants. If, then
Lemma 1. (See [8].) Let f and g be two non-constant meromorphic functions, and let $c_1$ , $c_2$ and $c_3$ be three non-zero constants. If $c_1f + c_2g = c_3$ , then
$$T(r, f) < \overline{N}\left(r, \frac{1}{f}\right) + \overline{N}\left(r, \frac{1}{g}\right) + \overline{N}(r, f) + S(r, f).$$
Lemma 2
Lemma 2. (See [6, 2].) Let be linearly independent meromorphic functions satisfying. Then for we have where D denotes the Wronskain and…
Lemma 2. (See [6, 2].) Let $f_1, f_2, \ldots, f_m$ be linearly independent meromorphic functions satisfying $\sum_{i=1}^m f_i = 1$ . Then for $k = 1, 2, \ldots, m$ we have
$$T(r, f_k) < \sum_{j=1}^m N\left(r, \frac{1}{f_j}\right) + N(r, f_k) + N(r, D) - \sum_{j=1}^m N(r, f_j)$$
$$-N\left(r, \frac{1}{D}\right) + o(T(r)) \quad (r \notin E),$$
where D denotes the Wronskain
$$D = \begin{vmatrix} f_1 & f_2 & \cdots & f_m \\ f'_1 & f'_2 & \cdots & f'_m \\ \vdots & \vdots & \vdots & \vdots \\ f_1^{(m-1)} & f_2^{(m-1)} & \cdots & f_m^{(m-1)} \end{vmatrix}$$
and T(r) denotes the maximum of $T(r, f_j)$ , j = 1, 2, ..., m.
Lemma 3
Lemma 3. (See [9].) Let, and be three meromorphic functions satisfying, and let, and. If, and are linearly independent, then, and are…
Lemma 3. (See [9].) Let $f_1$ , $f_2$ and $f_3$ be three meromorphic functions satisfying $\sum_{j=1}^{3} f_j = 1$ , and let $g_1 = -f_3/f_2$ , $g_2 = 1/f_2$ and $g_3 = -f_1/f_2$ . If $f_1$ , $f_2$ and $f_3$ are linearly independent, then $g_1$ , $g_2$ and $g_3$ are linearly independent.
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