Nevanlinna theory and its applications

(A)唯一性定理。作为Nevanlinna唯一性定理的推广，我们得到了如下结果。定理1。设f_1和f_2为复平面上的两个超越亚纯函数，在复平面上共享5个小亚纯函数a_1，…， a_5，使得a_j * *(j = 1,2,3,4)。(I)如果存在（1,2,3,4,5）的一个排列（p_1, p_2, p_3, p_4, p_5）使得交叉比[p_1， p_2, p_3， p_4]是一个常数或（II）如果存在一个j（1<小于或等于> j<小于或等于> 5）使得δ (a_j, f_j) > 15/17，则f_1 f_2。(B)全纯曲线的值分布。设f = f_1，…， f_<n+1>]是一条从C到P^n (C)的超越非简并全纯曲线。(1)定义了f的渐近点，并将其分为第一类和第二类。定理2。若f的低阶λ是有限的，则不同于一般位置的第一类渐近点的个数N满足以下条件：当λ <1/2_n时N<小于等于> N；如果1/2n<小于或等于> <1；N< <或等于>2_<nlambda>如果1<或等于这是对著名的Ahlfors边界点定理的推广。(2)改进了全纯曲线的第二个基本定理——缺陷关系，并应用它得到了具有最大缺陷和的全纯曲线的缺陷的一些结果。定理3。如果_SIGMA^q_<j=1> δ (a_j, f) = 2N - n + 1，如果ω < 1，则在{a_I，…， a_q}使得(a, f) = 1。(C)常微分方程的应用。例如，我们得到以下结果。定理4。设T(A)是复平面上微分方程（^1）^2 = A(z)（^2 - 1）的超越亚纯解的集合，其中A(z)是有理数。(a) T(a) =; (b); (c)(b) # <s:1> CT(A) 2；(c) # CT(A) =不可数。

(A) Unicity Theorem. As a generalizaion of the Unicity Theorem of Nevanlinna, we obtained the following result.Theorem 1. Let f_1 and f_2 be two transcendental meromorphic functions in te complex plane sharing 5 small merornorphic functions in te complex plane a_1, ..., a_5 , such that a_j * *(j = 1,2,3,4).(I) If there is a permutation (p_1, p_2, p_3, p_4, p_5) of (1,2,3,4,5) such that the cross ratio [p_1, p_2, p_3, p_4] is a constant or(II) If there is a j (1<less than or equal> j<less than or equal> 5) such that delta(a_j, f_j) > 15/17, then f_1 f_2.(B) Value distribution of holomorphic curves. Let f = f_1, ..., f_<n+1>] be a transcendental and non- degenerate holomorphic curve from C into P^n (C).(1) We defined an asymptotic spot for f and clssified it into two types , the first kind and the second. Theorem 2. If the lower order lambda of f is finite, the number N of asymptotic spots different in general position and of first kind satisfies the following :N<less than or equal>n if lambda<1/2_n ; N<less than or equal> 2_n - 1 if 1/2n<less than or equal> lambda<1 ; N<less than or equal>2_<nlambda> if 1<less than or equaThis is a generalization of the famous boundary point theorem of Ahlfors.(2) We ameliorated the second fundamental theorem, the defect relation for holomorphic curves and by applying them we obtained some results on defects of holomporphic curves with maximal deficiency sum.Theorem 3. If _SIGMA^q_<j=1>delta(a_j, f) = 2N - n + 1 and if OMEGA < 1, then there are at least [(N - n)(n - 1)/n] + 1 vectors a in {a_I, ..., a_q} such that delta(a, f) = 1.(C) Applications to ordinary differential equation. For example we obtained the following result.Theorem 4. Let T(A) be the set of transcendental meromorphic solutions of the differential equation(omega^1) ^2 = A(z)(omega^2 - 1),where A(z) is rational, in the complex plane. Then one of the following (a), (b) and (c) holds :(a) T(A) = phi ; (b) #ﾟCT(A) 2 ; (c) #ﾟCT(A) = uncountable.