Nevanlinna theory and its applications

(A)唯一性定理。定理1.设f_1和f_2是TE复平面上的两个超越亚纯函数，在TE复平面a_1，…，a_5中分担5个小的亚纯函数，使得a_j**(j=1，2，3，4)(I)(p_1，p_2，p_3，p_4，p_5)有(1，2，3，4，5)的一个排列(p_1，p_2，p_3，p_4，p_5)使得[p_1，p_2，p_3，p_1，p_2，p_4，p_5]的交叉比[p_1，p_2，p_3，.P_4]是一个常量或(Ii)如果存在j(1&lt；小于或等于&gt；j&lt；小于或等于&gt；5)使得Delta(a_j，f_j)&gt；15/17，然后f_1 f_2。(B)全纯曲线的值分布。设f=f_1，…，f_&lt；n+1&gt；是从C到P^n(C)的超越非退化全纯曲线。(1)定义了f的一个渐近点，并将其分为两类：第一类和第二类。定理2.如果f的低阶lambda是有限的，则在一般位置不同的第一类渐近点的个数N满足以下条件：N&lt；小于等于-gt；n如果lambda；lt；1/2_n；N&lt；小于或等于&gt；2_n-1，如果1/2n&lt；小于或等于&gt；这是Ahlfors著名的边界点定理的推广。(2)改进了第二个基本定理--全纯曲线的亏量关系，并应用它们得到了关于亏量和最大的全纯曲线的亏量的一些结果。1，则在{ai，…，a_q}中至少存在[(N-n)(n-1)/n]+1个向量a，使得Delta(a，f)=1。(C)应用于常微分方程。例如，我们得到了如下结果。定理4.设T(A)是复平面上微分方程(omega^1)^2=A(Z)(omega^2-1)的超越亚纯解集，其中A(Z)是有理的。则下列(A)、(B)和(C)之一成立：(A)T(A)=Phi；(B)#ﾟCT(A)2；(C)#ﾟ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.