亚纯函数正规族和值分布理论的一些新结果
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摘要
本文主要研究亚纯函数值分布和正规族理论,得到了一些新的结果,这些结果对原来的定理做了较大的改进.
首先,在第二章中间我们继续研究Picard型定理,得到了一个关于例外函数的Picard型定理,证明了:设f(z)是定义在复平面C上的超越亚纯函数,零点重数至少为k+1,极点重数至少为2,这里k≥2是整数.再设a(z) = P(z)exp(Q(z)) (?) 0,满足(?)=∞,这里P和Q是多项式,那么f~((k))(z) - a(z)在C上有无穷多个零点.
而在第三章,我们主要研究一类全纯函数的正规性,得到了两个主要定理,分别是:(1).设F是一族定义在区域D (?) C上的全纯函数.再设k≥2是一个整数,h(z)在D内解析,其所有零点的重数至多为k-1.假如对于任意的f∈F,
(a) f(z) = 0 (?) f'(z)=h(z);
(b) f'(z) = h(z) (?) |f~((k))(z)|≤c,这里c是常数;
(c) f(z)与h(z)在D内无公共零点,那么F在D上正规.
(2).设F是一族定义在区域D (?) C上的全纯函数,零点重数至少为k这里k(≠2)是一个整数.再设h(z)在D内解析,并且只有简单零点.假如对于任意的f∈F,
(a) f(z)=0 (?) f~((k))(z)=h(z);
(b) f~((k))(z) = h(z) (?) f~((k+1))(z) = 0;
(c) f(z)与h(z)在D内无公共零点,那么F在D上正规.
在第四章中,我们首先给出了一个拟正规定则,证明了:设D (?) C是单连通区域,{h_n}是定义在D上的一族全纯函数,满足h_n在D上内闭一致收敛到H'= cz~d,这里H在D内全纯且有H'≠0,∞, z∈D.设{f_n}是定义在D上的一族亚纯函数,并且对于每个n有,
(i) f_n的所有零点的重数至少为k+1,
(ii) f_n~((k))(z)≠h_n(z),z∈D,.那么{f_n}在D上是拟正规的,并且其拟正规的阶为|d + 1|.此外,如果{f_n}的每个子列在点Z_0∈D处均不正规,那么f_n~((k-1))(z)在D\{z_0}上按照球面距离内闭一致收敛于H(z) - H(z_0)且存在δ>0,使得对于所有的n有S(Δ(z_0,δ),f_n)≤k + 1.然后,将这个结果应用于值分布理论中,得到了如下的定理:f是定义在复平面C上的超越亚纯函数,其所有的零点除了有限多个以外的重级至少为k+1,并且设R(?)0是有理函数.那么f~((k))-R有无限多个零点.推广了庞学诚,S.Nevo和L.Zalcman的相关结果.最后,在第五章中,我们提出了一些未解决的问题.
In this paper, we mainly study the value distribution and normal family theory for meromorphic functions, and get some new results for them. These results deeply improved the former theorems.
In Chapter 2, we get a new Picard type theorem concerning omitted function, and prove the following result: Let f(z) be a transcendental meromorphic function on C, all of whose zeros have multiplicity at least k + 1, and all of whose poles are multiple, where k≥2 is an integer. Let the function a(z) = P(z)exp(Q(z)) (?) 0, where P and Q are polynomials such that (?)=∞. Then the function f~((k))(z) - a(z) has infinitely many zeros.
In Chapter 3, we study a new kind of holomorphic functions, and get the following two results: (a). Let F be a family of functions holomorphic on a domain D (?) C. Let k≥2 be an integer and let h(z) be an holomorphic function on D, all of whose zeros have multiplicity at most k-1, such that h(z) has no common zeros with any f∈F. Assume also that the following two conditions hold for every f∈F
(a) f(z) = 0 (?) f'(z)=h(z) and
(b) f'(z) = h(z) (?) |f~((k))(z)|≤c, where c is a constant. Then F is normal on D.
(b). Let F be a family of functions holomorphic on a domain D (?) C. Let k≠2 be an integer and let h(z) be an holomorphic function on D, all of zeros are simple, such that h(z) has no common zeros with any f∈F. Assume also that the following two conditions hold for every f∈F
(a) f(z) = 0 (?) f~((k))(z)=h(z) and
(b) f~((k))(z) = h(z) (?) f~((k+1))(z) = 0, where c is a constant. Then F is normal on D.
In Chapter 4, at first, we obtain a criterion for quasinormal families, and proved: Let D (?) C be a simple connected domain, and {h_n} be a sequence of holomorphic functions on D, such that h_n (?) H'= cz~d on D, where H is holomorphic on D and H'≠0,∞, z∈D. Let {f_n} be a sequence of meromorphic functions on D, all of whose zeros have multiplicity at least k + 1, such that f_n~((k))(z)≠h_n(z) for all n and all z∈D, then {f_n} is quasinormal of order |d + 1| on D. Moreover, if no subsequence of [f_n] is normal at z_0∈D, then f_n~((k-1))(z) (?) H(z) - H(z_0) on D\{z_0}, and for all n, S(Δ(z_0,δ),f_n)≤k + 1. Then, we use this result to get a Picard type theorem, and proved: Let f be a transcendental meromorphic function on C, all of whose zeros have multiplicity at least k + 1, and let R (?) 0 be a rational function, then f~((k)) - R has infinitely many zeros. This improved X.C.Pang, S.Nevo and L.Zalcman's result.
In Chapter 5, we give some unknown problems.
首先,在第二章中间我们继续研究Picard型定理,得到了一个关于例外函数的Picard型定理,证明了:设f(z)是定义在复平面C上的超越亚纯函数,零点重数至少为k+1,极点重数至少为2,这里k≥2是整数.再设a(z) = P(z)exp(Q(z)) (?) 0,满足(?)=∞,这里P和Q是多项式,那么f~((k))(z) - a(z)在C上有无穷多个零点.
而在第三章,我们主要研究一类全纯函数的正规性,得到了两个主要定理,分别是:(1).设F是一族定义在区域D (?) C上的全纯函数.再设k≥2是一个整数,h(z)在D内解析,其所有零点的重数至多为k-1.假如对于任意的f∈F,
(a) f(z) = 0 (?) f'(z)=h(z);
(b) f'(z) = h(z) (?) |f~((k))(z)|≤c,这里c是常数;
(c) f(z)与h(z)在D内无公共零点,那么F在D上正规.
(2).设F是一族定义在区域D (?) C上的全纯函数,零点重数至少为k这里k(≠2)是一个整数.再设h(z)在D内解析,并且只有简单零点.假如对于任意的f∈F,
(a) f(z)=0 (?) f~((k))(z)=h(z);
(b) f~((k))(z) = h(z) (?) f~((k+1))(z) = 0;
(c) f(z)与h(z)在D内无公共零点,那么F在D上正规.
在第四章中,我们首先给出了一个拟正规定则,证明了:设D (?) C是单连通区域,{h_n}是定义在D上的一族全纯函数,满足h_n在D上内闭一致收敛到H'= cz~d,这里H在D内全纯且有H'≠0,∞, z∈D.设{f_n}是定义在D上的一族亚纯函数,并且对于每个n有,
(i) f_n的所有零点的重数至少为k+1,
(ii) f_n~((k))(z)≠h_n(z),z∈D,.那么{f_n}在D上是拟正规的,并且其拟正规的阶为|d + 1|.此外,如果{f_n}的每个子列在点Z_0∈D处均不正规,那么f_n~((k-1))(z)在D\{z_0}上按照球面距离内闭一致收敛于H(z) - H(z_0)且存在δ>0,使得对于所有的n有S(Δ(z_0,δ),f_n)≤k + 1.然后,将这个结果应用于值分布理论中,得到了如下的定理:f是定义在复平面C上的超越亚纯函数,其所有的零点除了有限多个以外的重级至少为k+1,并且设R(?)0是有理函数.那么f~((k))-R有无限多个零点.推广了庞学诚,S.Nevo和L.Zalcman的相关结果.最后,在第五章中,我们提出了一些未解决的问题.
In this paper, we mainly study the value distribution and normal family theory for meromorphic functions, and get some new results for them. These results deeply improved the former theorems.
In Chapter 2, we get a new Picard type theorem concerning omitted function, and prove the following result: Let f(z) be a transcendental meromorphic function on C, all of whose zeros have multiplicity at least k + 1, and all of whose poles are multiple, where k≥2 is an integer. Let the function a(z) = P(z)exp(Q(z)) (?) 0, where P and Q are polynomials such that (?)=∞. Then the function f~((k))(z) - a(z) has infinitely many zeros.
In Chapter 3, we study a new kind of holomorphic functions, and get the following two results: (a). Let F be a family of functions holomorphic on a domain D (?) C. Let k≥2 be an integer and let h(z) be an holomorphic function on D, all of whose zeros have multiplicity at most k-1, such that h(z) has no common zeros with any f∈F. Assume also that the following two conditions hold for every f∈F
(a) f(z) = 0 (?) f'(z)=h(z) and
(b) f'(z) = h(z) (?) |f~((k))(z)|≤c, where c is a constant. Then F is normal on D.
(b). Let F be a family of functions holomorphic on a domain D (?) C. Let k≠2 be an integer and let h(z) be an holomorphic function on D, all of zeros are simple, such that h(z) has no common zeros with any f∈F. Assume also that the following two conditions hold for every f∈F
(a) f(z) = 0 (?) f~((k))(z)=h(z) and
(b) f~((k))(z) = h(z) (?) f~((k+1))(z) = 0, where c is a constant. Then F is normal on D.
In Chapter 4, at first, we obtain a criterion for quasinormal families, and proved: Let D (?) C be a simple connected domain, and {h_n} be a sequence of holomorphic functions on D, such that h_n (?) H'= cz~d on D, where H is holomorphic on D and H'≠0,∞, z∈D. Let {f_n} be a sequence of meromorphic functions on D, all of whose zeros have multiplicity at least k + 1, such that f_n~((k))(z)≠h_n(z) for all n and all z∈D, then {f_n} is quasinormal of order |d + 1| on D. Moreover, if no subsequence of [f_n] is normal at z_0∈D, then f_n~((k-1))(z) (?) H(z) - H(z_0) on D\{z_0}, and for all n, S(Δ(z_0,δ),f_n)≤k + 1. Then, we use this result to get a Picard type theorem, and proved: Let f be a transcendental meromorphic function on C, all of whose zeros have multiplicity at least k + 1, and let R (?) 0 be a rational function, then f~((k)) - R has infinitely many zeros. This improved X.C.Pang, S.Nevo and L.Zalcman's result.
In Chapter 5, we give some unknown problems.
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