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 \ w +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 \ w 11 -\-aw9 +b = 0}, wiere a and b are two non-zero constants such that b² ^ —2² 99 (a/ll) . Tien for any two nonconstant meromorphic /unctions / and g, the conditions Ef(S) = Eg(S) and JS/({oo}) = Eg({oo}) imply f=9-

Lemma 1. (see [6]). Let f and g be two nonconstant meromorpbic functions, and let c\ , c2 and C3 be three nonzero 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. (see [4]). Let /1 , f2, ..., fn be hnearly independent meromorphic functions satisfying $$\sum_{j=1}^n f_j = 1.$$ Tien for k — 1, 2, ... , n we have $$T(r, f_k) < \sum_{j=1}^n N\left(r, \frac{1}{f_j}\right) + N(r, f_k) + N(r, D) - \sum_{j=1}^n N(r, f_j)$$ $$-N\left(r, \frac{1}{D}\right) + o(T(r)) \quad (r \notin E),$$ where D denotes the WronsMan of the functions /1 , / ² , ... , / " and T(r) denotes the maximum of T(r, fj),j = l,2,...,n.

Lemma 3. (see [5]). Let f be a nonconstant meromorpbic function, and let P(f) be a polynomial in f of the form $$P(f) = a_0 f^n + a_1 f^{n-1} + \ldots + a_{n-1} f + a_n,$$ where OQ (^ 0), OI, ... , on are constants. Then $$T(r, P(f)) = nT(r, f) + S(r, f).$$

Lemma 4. $f_1$ , $f_2$ and $f_3$ are linearly dependent. PROOF: Suppose that $f_1$ , $f_2$ and $f_3$ are linearly independent. Applying Lemma 2 to the functions $f_j$ (j = 1, 2, 3), from (8) and (9) we have (10) $$T(r, f_1) < \sum_{j=1}^{3} N\left(r, \frac{1}{f_j}\right) - N\left(r, \frac{1}{D}\right) + N(r, D) - N(r, f_2) - N(r, f_3) + S(r, f),$$ where (11) $$D = \begin{vmatrix} f_1 & f_2 & f_3 \\ f'_1 & f'_2 & f'_3 \\ f''_1 & f''_2 & f''_3 \end{vmatrix}.$$ From (5), (6) and (7) we have $$\sum_{j=1}^{3} N\left(r, \frac{1}{f_{j}}\right) = (n-m)N\left(r, \frac{1}{f}\right) + N\left(r, \frac{1}{f^{m}+a}\right) + (n-m)N\left(r, \frac{1}{g}\right) + N\left(r, \frac{1}{g^{m}+a}\right).$$ By looking at the zeros of f and g, from (5), (6), (7) and (11) we see that $$(13) \ \ N\left(r,\frac{1}{D}\right)\geqslant (n-m)N\left(r,\frac{1}{f}\right)-2\overline{N}\left(r,\frac{1}{f}\right)+(n-m)N\left(r,\frac{1}{g}\right)-2\overline{N}\left(r,\frac{1}{g}\right).$$ From (8) and (11) we get (14) $$D = \begin{vmatrix} f_2' & f_3' \\ f_2'' & f_3'' \end{vmatrix}.$$ Since $f_2$ is entire, from (7) and (14) we have (15) $$N(r, D) - N(r, f_2) - N(r, f_3) \leq N(r, f_3'') - N(r, f_3)$$ $$= 2\overline{N}(r, f_3) = 2\overline{N}(r, q) = 2\overline{N}(r, f).$$ From Lemma 3, (5), (10), (12), (13) and (15) we deduce $$nT(r, f) < 2\overline{N}\left(r, \frac{1}{f}\right) + N\left(r, \frac{1}{f^m + a}\right) + 2\overline{N}\left(r, \frac{1}{g}\right)$$ $$+ N\left(r, \frac{1}{g^m + a}\right) + 2\overline{N}(r, f) + S(r, f)$$ $$< (4 + m)T(r, f) + (2 + m)T(r, g) + S(r, f).$$ From this and (1) we obtain (16) $$nT(r, f) < \left(2m + 6 + \frac{2(2+m)}{n-1}\right)T(r, f) + S(r, f).$$ Since n ^ 2?n + 7, (16) is a contradiction, which proves Lemma 4. D Next we proceed to prove Theorem 2. By Lemma 4 we know that f\ , /z and fa are linearly dependent. Then there exist three constants ci, C2 and C3, at least one of which is not zero, such that (17) $$c_1 f_1 + c_2 f_2 + c_3 f_3 = 0.$$ If Ci = 0, from (17) we have c2 ^ 0, C3 ^ 0 and $$f_3=-\frac{c_2}{c_3}f_2.$$ Hence, from (6) and (7) we obtain $$g^n + ag^{n-m} = -bc_2/c_3,$$ which is impossible. Thus C\ ^ 0 and $$f_1 = -\frac{c_2}{c_1} f_2 - \frac{c_3}{c_1} f_3.$$ Now combining (8) and (18) we get (19) $$\left(1 - \frac{c_2}{c_1}\right) f_2 + \left(1 - \frac{c_3}{c_1}\right) f_3 = 1.$$ We discuss the following three cases. CASE I. Assume cx ^ c2 and ci ^ c ³ . From (6), (7) and (19) we have (20) $$-\frac{1}{b}\left(1-\frac{c_3}{c_1}\right)g^{n-m}(g^m+a)+c^{-h}=1-\frac{c_2}{c_1}$$ From (20) we know that g is a entire function. By Lemma 1, Lemma 3 and (20) we obtain $$nT(r, g) < \overline{N}\left(r, \frac{1}{g^{n-m}(g^m + a)}\right) + S(r, g)$$ $$= \overline{N}\left(r, \frac{1}{g}\right) + \overline{N}\left(r, \frac{1}{g^m + a}\right) + S(r, g)$$ $$< (1 + m)T(r, g) + S(r, g),$$ which is impossible. CASE II. Assume ci = c2. From (19) we have cx ^ c3 and $$f_3 = \frac{c_1}{c_1 - c_3}.$$ From (7) and (21) we get (22) $$g^{n-m}(g^m+a)=\frac{bc_1}{c_1-c_3}e^{-h}.$$ Let a\, at, ..., am be the roots of equation wm + a — 0. From (22) we know that 00, 0, Oi, 02, ..., om are Picard exceptional values of g, which is impossible. CASE III. Assume ci = c3. From (19) we have ci ^ c2 and $$f_2=\frac{c_1}{c_1-c_2}$$ that is (23) $$e^h = \frac{c_1}{c_1 - c_2}.$$ From (5), (7), (8) and (23) we get $$-f^{n-m}(f^m+a)+\frac{c_1}{c_1-c_2}g^{n-m}(g^m+a)=\frac{bc_2}{c_2-c_1}.$$ If C2 ^ 0, by Lemma 1 and Lemma 3, we have from (24), $$nT(r, f) < \overline{N}\left(r, \frac{1}{f^{n-m}(f^m + a)}\right) + \overline{N}\left(r, \frac{1}{g^{n-m}(g^m + a)}\right) + \overline{N}(r, f) + S(r, f)$$ $$= \overline{N}\left(r, \frac{1}{f}\right) + \overline{N}\left(r, \frac{1}{f^m + a}\right) + \overline{N}\left(r, \frac{1}{g}\right)$$ $$+ \overline{N}\left(r, \frac{1}{g^m + a}\right) + \overline{N}(r, f) + S(r, f)$$ $$< (2 + m)T(r, f) + (1 + m)T(r, g) + S(r, f).$$ From this and (1) we obtain $$nT(r, f) < \left(3 + 2m + \frac{2(1+m)}{n-1}\right)T(r, f) + S(r, f),$$ which is impossible. Thus c% = 0. From (24) we deduce (25) $$f^{n} - g^{n} = -a(f^{n-m} - g^{n-m}).$$ H f ^ g, from (25) we obtain (26) $$g^{m} = -\frac{a(H-v)(H-v^{2})\dots(H-v^{n-m-1})}{(H-u)(H-u^{2})\dots(H-u^{n-1})},$$ where H = f/g, u = exp((27ri)/n) and t; = exp ((2iri)/(n — m)). From (26) we know that H is a nonconstant meromorphic function. Since n and m have no common factors, we have it\' j£ v (j = 1, 2, ... , n — 1; k = 1, 2, ... , n — m — 1). Suppose that z,- is a zero of if — u3 of order p^-. From (26) we have pj ^ m* ^ 2. Thus (27) $$\overline{N}\left(r, \frac{1}{H-u^j}\right) \leqslant \frac{1}{m}N\left(r, \frac{1}{H-u^j}\right) \leqslant \frac{1}{2}T(r, H) + O(1).$$ By the second fundamental theorem, from (27) we obtain $$\begin{split} &(n-3)T(r,\,H) < \sum_{j=1}^{n-1} \overline{N}\bigg(r,\,\frac{1}{H-u^j}\bigg) + S(r,\,H) \ &< \frac{n-1}{2}T(r,\,H) + S(r,\,H), \end{split}$$ which is impossible. Hence, f = g. This completes the proof of Theorem 2.

Theorem 3. Let $S = \{w \mid w^n + aw^{n-1} + b = 0\}$ , where n > 8, and a and b are two nonzero constants such that the algebraic equation $w^n + aw^{n-1} + b = 0$ has no multiple roots. Suppose that f and g are two distinct nonconstant meromorphic functions satisfying $E_f(S) = E_g(S)$ and $E_f(\{\infty\}) = E_g(\{\infty\})$ . Then $$f = -\frac{aH(H^{n-1}-1)}{H^n-1}$$ and $g = -\frac{a(H^{n-1}-1)}{H^n-1}$ , where H is a nonconstant meromorphic function. PROOF: Proceeding as in the proof of Theorem 2, we have (28) $$f^{n} - g^{n} = -a(f^{n-1} - g^{n-1}).$$ Noting $f \not\equiv g$ , from (28) we obtain (29) $$g = -\frac{a(H^{n-1}-1)}{H^n-1},$$ where H = f/g. From (29) we know that H is a nonconstant meromorphic function. Thus, from (29) we have $$f = -\frac{aH\left(H^{n-1} - 1\right)}{H^n - 1}.$$ This completes the proof of Theorem 3.