非齐次线性微分方程亚纯解的零点和极点分布

On the Distribution of Zeros and Poles of Meromorphic Solutions of Non-homogeneous Linear Differential Equations

关键词:  线性微分方程  亚纯解  零点收敛指数  极点收敛指数

Key words:  linear differential equation  meromorphic solution  zero-exponent of convergence  pole-exponent of convergence

摘要:

以d(g)表示亚纯函数g(z)的增长级;λ(g),λ(1/g)分别表示g(z)的零点、极点序列收敛指数;λ(g),λ(1/g)分别表示g(z)的不同零点、极点序列收敛指数.定理1设B_j(z)(j=0,1,…,k-1;k≥1)和F(z)为亚纯函数且满足d(F)>max_{j=0,1,…,k-1}{d(B_j)},又设f(z)为f^(k)+B_k-1(z)f^(k-1)+…+Bo(z)f=F(z)的一个亚纯函数解.则:(a)若d(f)>d(F),则,max{λ(f),λ(1/f)}.(b)若d(F)为有穷非正整数,则max{λ(f),λ(1/f)}≥max{λ(F),λ(1/F)}.定理2设R(z),Q(z)为非零亚纯函数,且满足max{d(Bo),…,d(B_k-1),d(Q),λ(R)}Y).又设f(z)为f(k)+B_k-1(z)f^(k-1)+…+Bo(z)f=R(z)+Q(z)≡Y(z)的亚纯函数解,则:(a)若d(Y)<∞,则,max{λ(f),λ(1/f)}≥max{λ(f),λ(1/Y))}.(b)若d(Y)=∞.则max{λ(f),λ(1/f)}=∞.

Abstract:

Suppose, d(g) is the order of growth of meromorphic function g(z);λ(g),λ(1/g) are zero-exponent and pole-exponent of convergence of g(z), respectively;λ(g),λ(1/g),are different zero-exponent and pole-exponent of convergence of g(z),respectively. Theorem 1 Suppose B_j(z)(j=0,1,…,k-1;k ≥ 1) and F(z) are meromorphic functions, and satisfy d(F)> max_j=0,1,k-1{d(B_j)}, and suppose f(z) is a meromorphic solution of f^(k)+B_k-1(z)f^(k-1)+…+Bo(z)f=F(z). Then:(a) if d(f) >d(F), then, max {λ(f),λ(1/f)}=d(f). (b) if d(F) is finite non-positive integer, then, max {λ(f),λ(1/f)} ≥ max {λ(F),λ(1/F)}. Theorem 2 Suppose R(z),Q(z) are non-zero meromorphic solutions, and satisfy max {d(Bo),…,d(B_k-1),d(Q), λ(R)} Y), and suppose f(z) is meromorphic solution of f^(k)>+B_k-1, (z)f^(k-1)+…+Bo(z)f=R(z)+Q(z)=Y(z). Then:(a) if d(Y)<∞, then, max{λ(f),λ(1/f)} ≥ max{λ(Y),λ(1/Y)}. (b) if d(Y)=∞, then, max{λ(f),λ(1/f)}=∞.