Nevanlinna theory and its applications

点击翻译按钮获取中文摘要

(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.