整函数与其微分多项式的唯一性理论
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摘要
二十世纪二十年代，芬兰数学家R．Nevanlinna引进了亚纯函数的特征函数，并建立了两个基本定理，从而创立了Nevanlinna值分布理论．他所创立的这一理论是二十世纪最重大的数学成就之一，不仅奠定了现代亚纯函数理论的基础，而且对数学的许多分支的发展，交叉和融合产生了重大而深远的影响．随着Nevanlinna理论自身的不断发展，以它为主要研究工具的亚纯函数唯一性理论取得了蓬勃的发展．
本文主要介绍作者在扈培础教授的精心指导下，就单或多变量整函数涉及微分多项式的唯一性问题所做的一些研究，得到了一些结果．全文共分为三章．
在第一章中，作者扼要介绍了本文的研究背景，Nevanlinna基本理论中的常用记号，并叙述了亚纯幽数唯一性理论中的一些基本概念和结果．
在第二章中，作者研究了复数域C上的非常数整函数与其线性微分多项式具有两个公共值的唯一性问题，推广了Bernstein-Chang-Li及Li-Yang的结果，主要结论如下：
定理1：设f为非常数整函数，令L(f)为f的线性微分多项式，形式如下其中b_n((?)0)．b_(n-1)，…,b_0,b_(-1)为f的亚纯小丽数．若f与L(f)以两个判别的有穷复数a_1，a_2为其IM公共值，且L(f)所有单重的a_1值点均为f的单重a_1值点，则f与L(f)分担a_1CM，并且或者f≡L(f)，或者它们满足下面的形式其中α是非常数整函数．
推论2：在定理1的条件下，如果f与L(f)分担两个判别有穷值a_1,a_2IM,且L(f)的所有单a_1和a_2值点都是f的单a_1和a_2值点，那么f与L(f)分担a_1和a_2CM，且有f≡L(f)．
在第三章中，作者研究了C~n上的整函数与其线性微分多项式分担一个公共小函数的唯一性问题，得到的主要结论如下：
定理3：设f是定义在C~n上的超越整函数，L(f)是其线性微分多项式，形式如下：其中a_(l_1,l_2,...l_n)∈C为常数，且至少有一个a_(l_1,l_2,...l_n)≠0(l_1+l_2+...+l_n=k)，a(z)是f的亚纯小函数且a(z)(?)0，∞,如果f与L(f)分担a(z)CM,且δ(0，f)>1/2，那么f≡L(f)．
In 1920s, R.Nevanlinna introduced the characteristic functions of meromorphirfunctions, proved two fundamental theorems.and established so-called value distribute theory or called Nevanlinna theory. This theory is one of the most importantachievements in the 20th century praised by H.weyl. It plays basic roles for modern researches of meromorphic function theory and has a very important inflnence on the development and syncretism of many mathematical branches. Associatedto the development of the Nevanlinna theory itself, some new mathematical branches have appeared based on main methods and results from Nevanlinna theory.In the late 1920s, R.Nevanlinna also studied the conditions which determine completely a meromorphic function, and obtained three celebrated uniqueness theoremsfor meromorphic functions, which are usually called Nevaulinna's five-value thenrcm, four-value theorem and threc-valuc theorem respectively. This launched the investigation of uniqueness theory of meromorphic functions.
The present thesis is a part of the author's research work on the uniqueness problems of entire functions that share values or small functions with their linear differential polynomials over C or C~n. It consists of three chapters.
In chapter 1. we briefly introduce the background of this thesis, which containssome fundamental results and notations of Nevanlinna theory.
In chapter 2, we improve the results of Bernstcin-Chang-Li and Li-Yang by studying the uniqueness problem of entire functions that share two values with their linear differential polynomials. The main result is the following:
Theorem 1. Let f be a non-constant entire function, and let L(f) be a linear differential polynomial in f of the following formwhore the coefficients b_n((?)0), b_(n-1),…., b_0, b_(-1) are small meromorphic functions of f. Assume that f and L(f) share two distinct finite values a_1, a_2 IM, and all the simple a_1-points of L(f) are simple a_1-points of f. Then f and L(f) share the value a_1 CM, and So either f≡L(f), or they have the following expressionsandwhereαis a non-constant entire function.
Corollary 2. Under the conditions of Theorem 1. if we furthermore assume that f and L(f) share two distinct finite valuesα_1,α_2 IM, and all the simpleα_1-points andα_2-points of L(J) are simpleα_1-points andα_2-points of f respectively, then f and L(f) share the valuesα_1.α_2 CM and f≡L(f).
In Chapter 3, we makes the research of uniqueness problem of entire functionson C~n that share one small function with their linear differential polynomials. The main result is the following:
Theorem 3. Let f be a transcendental entire function on C~n, let L(f) be a linear differential polynomial in f of the following form:where a(l_1,l_2,…l_n)∈C are constants and at least one of a(l_1,l_2,…l_n)≠0 (l_1+ l_2+…+ l_n = k), and let a(z) be a small meromorphic function of f such that a(z)(?) 0,∞. If f and L(f) share a(z) CM, andδ(0,f) > 1/2, then f(?) L(f).
本文主要介绍作者在扈培础教授的精心指导下，就单或多变量整函数涉及微分多项式的唯一性问题所做的一些研究，得到了一些结果．全文共分为三章．
在第一章中，作者扼要介绍了本文的研究背景，Nevanlinna基本理论中的常用记号，并叙述了亚纯幽数唯一性理论中的一些基本概念和结果．
在第二章中，作者研究了复数域C上的非常数整函数与其线性微分多项式具有两个公共值的唯一性问题，推广了Bernstein-Chang-Li及Li-Yang的结果，主要结论如下：
定理1：设f为非常数整函数，令L(f)为f的线性微分多项式，形式如下其中b_n((?)0)．b_(n-1)，…,b_0,b_(-1)为f的亚纯小丽数．若f与L(f)以两个判别的有穷复数a_1，a_2为其IM公共值，且L(f)所有单重的a_1值点均为f的单重a_1值点，则f与L(f)分担a_1CM，并且或者f≡L(f)，或者它们满足下面的形式其中α是非常数整函数．
推论2：在定理1的条件下，如果f与L(f)分担两个判别有穷值a_1,a_2IM,且L(f)的所有单a_1和a_2值点都是f的单a_1和a_2值点，那么f与L(f)分担a_1和a_2CM，且有f≡L(f)．
在第三章中，作者研究了C~n上的整函数与其线性微分多项式分担一个公共小函数的唯一性问题，得到的主要结论如下：
定理3：设f是定义在C~n上的超越整函数，L(f)是其线性微分多项式，形式如下：其中a_(l_1,l_2,...l_n)∈C为常数，且至少有一个a_(l_1,l_2,...l_n)≠0(l_1+l_2+...+l_n=k)，a(z)是f的亚纯小函数且a(z)(?)0，∞,如果f与L(f)分担a(z)CM,且δ(0，f)>1/2，那么f≡L(f)．
In 1920s, R.Nevanlinna introduced the characteristic functions of meromorphirfunctions, proved two fundamental theorems.and established so-called value distribute theory or called Nevanlinna theory. This theory is one of the most importantachievements in the 20th century praised by H.weyl. It plays basic roles for modern researches of meromorphic function theory and has a very important inflnence on the development and syncretism of many mathematical branches. Associatedto the development of the Nevanlinna theory itself, some new mathematical branches have appeared based on main methods and results from Nevanlinna theory.In the late 1920s, R.Nevanlinna also studied the conditions which determine completely a meromorphic function, and obtained three celebrated uniqueness theoremsfor meromorphic functions, which are usually called Nevaulinna's five-value thenrcm, four-value theorem and threc-valuc theorem respectively. This launched the investigation of uniqueness theory of meromorphic functions.
The present thesis is a part of the author's research work on the uniqueness problems of entire functions that share values or small functions with their linear differential polynomials over C or C~n. It consists of three chapters.
In chapter 1. we briefly introduce the background of this thesis, which containssome fundamental results and notations of Nevanlinna theory.
In chapter 2, we improve the results of Bernstcin-Chang-Li and Li-Yang by studying the uniqueness problem of entire functions that share two values with their linear differential polynomials. The main result is the following:
Theorem 1. Let f be a non-constant entire function, and let L(f) be a linear differential polynomial in f of the following formwhore the coefficients b_n((?)0), b_(n-1),…., b_0, b_(-1) are small meromorphic functions of f. Assume that f and L(f) share two distinct finite values a_1, a_2 IM, and all the simple a_1-points of L(f) are simple a_1-points of f. Then f and L(f) share the value a_1 CM, and So either f≡L(f), or they have the following expressionsandwhereαis a non-constant entire function.
Corollary 2. Under the conditions of Theorem 1. if we furthermore assume that f and L(f) share two distinct finite valuesα_1,α_2 IM, and all the simpleα_1-points andα_2-points of L(J) are simpleα_1-points andα_2-points of f respectively, then f and L(f) share the valuesα_1.α_2 CM and f≡L(f).
In Chapter 3, we makes the research of uniqueness problem of entire functionson C~n that share one small function with their linear differential polynomials. The main result is the following:
Theorem 3. Let f be a transcendental entire function on C~n, let L(f) be a linear differential polynomial in f of the following form:where a(l_1,l_2,…l_n)∈C are constants and at least one of a(l_1,l_2,…l_n)≠0 (l_1+ l_2+…+ l_n = k), and let a(z) be a small meromorphic function of f such that a(z)(?) 0,∞. If f and L(f) share a(z) CM, andδ(0,f) > 1/2, then f(?) L(f).
引文
[1] R.Nevanlinna, Le Theoreme de Picard-Borel et la theorie des fonctions meromorphes. Paris, 1929.
[2] W. K. Hayman, Meromorphic Functions. Clarendon Press, Oxford, 1964.
[3] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions. Science Press & Kluwer, Beijing, 2003.
[4] Hu P.Ch. and Li P. and C.C.Yang. Unicity of meromorphic mappings. Kulwer Academic Publishers, The Netherlnds, 2003
[5] L. A. Rubel and C. C. Yang. Values shared by an entire function and its derivative. Lecture Notes in Math.. 599, Springer-Verlag, New York, 1977.
[6] C. A. Bernstein. D. C. Chang and B. Q. Li, Shared values for meromorphic functions. Adv. Math. 115(1995), 201-220.
[7] C. A. Bernstein, D. C. Chang and B. Q. Li. The uniqueness problem and meromorphic solutions of partial differential equations. J. D'Anal. Math. 77(1999), 51-68.
[8] G. Frank and G. Weisscnborn. Mcromorphc Funktionen die mil einer ihrer Ableitungen Werte teilen. Complex Variables 7(1986), 33-43.
[9] G. Frank and X. H. Una, Differential polynomials that slum three finite values with their generating meromorphic function. Michigan Math. J. 46(1999), 175-186.
[10] G. G. Gundersen. Mcromorplnc functions that share finite values with their derivatives. J. Math. Anal. Appl. 75(1980), 441-446.
[11]G. G. Gundersen. Meromorphic functions that share two finite values with their derivatives. Pacific J. Math. 105(1983), 209-309.
[12] P. Li, Meromorphic functions sharing three values IM with its linear differentialpolynomials. Complex Variales 41(2000), 221-229.
[13] P. Li and C. C. Yang, When an entire function and its linear differential polynomail share two values. Illinois J. Math. 44(2000), 349-361.
[14] C. A. Bernstein. D. C. Chang and B. Q. Li. On uniqueness of entire, functions in C~n and their partial differential polynomials. Forum Math. 8(1996), 379-396.
[15] P. Li and C. C. Yang, Value sharing of an entire function and its derivatives.J. Math. Soc. Japan 51(1999), 781-800.
[16] P. Li and C. C. Yang , When a entire function and its linear differential polynomial share two values. Illinois J.Math. 44(2000) .221-229.
[17] P. Li and C. C. Yang ,Further results on meromorphic functions that share two values with their derivatives. J. Aust. Math. Soc. 78(2005), 91-102.
[18] R.Bn'ick , On entire function which share one value CM with their first derivative. Results in Math., 30(1996), 21-24.
[19] G.G.Guixlersen and L-Zh Yang, Entire, function that share one value with one or two of their derivatives. J. Math. Anal. Appl., 223(1998). 88-95.
[20] L-Zh Yang, Solution of a differential equation and its applications. Kodai Math.,)., 22(1999), 458-404.
[21] Kit-Wing Yu, On entire and meromorphic functions that share small functionswith their derivatives. J. Inequal. Pure and Appl.Math., 2003, 4(1).
[22] Liu.L.P. and Gu.Y.X. ,Uniqueness of mesomorphic functions that share one small function with their derivatives. Kodai math J., 27(2004), 272-279.
[23] Jin L. ,A unicity theorem for entire function of several complex variables. Chin. Ann. Math., 25B:4(2004). 483-492.
[2] W. K. Hayman, Meromorphic Functions. Clarendon Press, Oxford, 1964.
[3] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions. Science Press & Kluwer, Beijing, 2003.
[4] Hu P.Ch. and Li P. and C.C.Yang. Unicity of meromorphic mappings. Kulwer Academic Publishers, The Netherlnds, 2003
[5] L. A. Rubel and C. C. Yang. Values shared by an entire function and its derivative. Lecture Notes in Math.. 599, Springer-Verlag, New York, 1977.
[6] C. A. Bernstein. D. C. Chang and B. Q. Li, Shared values for meromorphic functions. Adv. Math. 115(1995), 201-220.
[7] C. A. Bernstein, D. C. Chang and B. Q. Li. The uniqueness problem and meromorphic solutions of partial differential equations. J. D'Anal. Math. 77(1999), 51-68.
[8] G. Frank and G. Weisscnborn. Mcromorphc Funktionen die mil einer ihrer Ableitungen Werte teilen. Complex Variables 7(1986), 33-43.
[9] G. Frank and X. H. Una, Differential polynomials that slum three finite values with their generating meromorphic function. Michigan Math. J. 46(1999), 175-186.
[10] G. G. Gundersen. Mcromorplnc functions that share finite values with their derivatives. J. Math. Anal. Appl. 75(1980), 441-446.
[11]G. G. Gundersen. Meromorphic functions that share two finite values with their derivatives. Pacific J. Math. 105(1983), 209-309.
[12] P. Li, Meromorphic functions sharing three values IM with its linear differentialpolynomials. Complex Variales 41(2000), 221-229.
[13] P. Li and C. C. Yang, When an entire function and its linear differential polynomail share two values. Illinois J. Math. 44(2000), 349-361.
[14] C. A. Bernstein. D. C. Chang and B. Q. Li. On uniqueness of entire, functions in C~n and their partial differential polynomials. Forum Math. 8(1996), 379-396.
[15] P. Li and C. C. Yang, Value sharing of an entire function and its derivatives.J. Math. Soc. Japan 51(1999), 781-800.
[16] P. Li and C. C. Yang , When a entire function and its linear differential polynomial share two values. Illinois J.Math. 44(2000) .221-229.
[17] P. Li and C. C. Yang ,Further results on meromorphic functions that share two values with their derivatives. J. Aust. Math. Soc. 78(2005), 91-102.
[18] R.Bn'ick , On entire function which share one value CM with their first derivative. Results in Math., 30(1996), 21-24.
[19] G.G.Guixlersen and L-Zh Yang, Entire, function that share one value with one or two of their derivatives. J. Math. Anal. Appl., 223(1998). 88-95.
[20] L-Zh Yang, Solution of a differential equation and its applications. Kodai Math.,)., 22(1999), 458-404.
[21] Kit-Wing Yu, On entire and meromorphic functions that share small functionswith their derivatives. J. Inequal. Pure and Appl.Math., 2003, 4(1).
[22] Liu.L.P. and Gu.Y.X. ,Uniqueness of mesomorphic functions that share one small function with their derivatives. Kodai math J., 27(2004), 272-279.
[23] Jin L. ,A unicity theorem for entire function of several complex variables. Chin. Ann. Math., 25B:4(2004). 483-492.