涉及分担值的亚纯函数的正规定则
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摘要
本文主要研究亚纯函数的正规性问题。正规性是单复变函数中的一个重要研究课题，国内外许多学者对此作出了大量卓有成效的研究工作。在前言中，我们对复变函数及其历史背景以及这些领域的研究成果作了一番综述。全文共分为三章。
在第一章中，我们给出本文所要用到的一些基础知识：亚纯函数值分布理论的方面的基础知识及一些常用的记号，复分析和正规族里的一些基本结果。
第二章讨论了亚纯函数关于它的微分多项式的几个正规定则，主要证明了定理2.1。
第三章讨论亚纯函数关于分担值的一些正规定则，主要证明了定理3.1，定理3.2和定理3.3。
In this paper we study normal families of meromorphic functions, which is an important subject in complex analysis. Much work was made on this respect. In the preface, we give a review about the historical background of complex analysis and research achievements in these fields. The present paper is divided into three parts.
In chapter one we state a few usual notations, definitions and basic results in value distribution theory and the theory of normal families of meromorphic function.
In chapter two, we discuss the normal families of meromorphic functions concerning differential polynomial and mainly prove Theorem 2.1.
In chapter three, we discuss the normal families of meromorphic functions concerning shared values and mainly prove Theorem 3.1, Theorem 3.2 and Theorem 3.3.
在第一章中，我们给出本文所要用到的一些基础知识：亚纯函数值分布理论的方面的基础知识及一些常用的记号，复分析和正规族里的一些基本结果。
第二章讨论了亚纯函数关于它的微分多项式的几个正规定则，主要证明了定理2.1。
第三章讨论亚纯函数关于分担值的一些正规定则，主要证明了定理3.1，定理3.2和定理3.3。
In this paper we study normal families of meromorphic functions, which is an important subject in complex analysis. Much work was made on this respect. In the preface, we give a review about the historical background of complex analysis and research achievements in these fields. The present paper is divided into three parts.
In chapter one we state a few usual notations, definitions and basic results in value distribution theory and the theory of normal families of meromorphic function.
In chapter two, we discuss the normal families of meromorphic functions concerning differential polynomial and mainly prove Theorem 2.1.
In chapter three, we discuss the normal families of meromorphic functions concerning shared values and mainly prove Theorem 3.1, Theorem 3.2 and Theorem 3.3.
引文
[1] W. K. Hayman, Meromorphic Functions, Clarendon Press Oxford, 1964.
[2] J. Schiff, Normal Families, Springer-Verlag, 1993.
[3] L. Yang, Value Distribution Theory, Springer-Verlag&Science Press, Berlin, 1993.
[4] P. Montel, Sur les families de fonctions analytiques qui admettent des valuers exceptionnelles dans un domaine, Ann. cole Norm. Sup. (3)29(1912), 487-535.
[5] D. C. Sun, The shared value criterion for normality, J. Wuhan Univ. Natur. Sci. Ed. 3(1994), 9-12.
[6] Y. X. Gu, Un critère de normalité de fonctions méromorphes, Sci. Sinica, Special Issue 1(1979), 267-274.
[7] M. L. Fang and L. Zalcman, A note on normality and shared values, J. Austral Math. Soc, 76(2004), 141-150.
[8] L. Yang, A general criterion of normality, Acta Math. Sinica (New series) 1 (1985), 181-193.
[9] 朱经浩，亚纯函数的一个总的正规定则，科学通报，3(1986)，174-177。
[10] 王跃飞和方明亮，涉及零点重数的亚纯函数的值分布，数学学报，4(1998)，743-748．
[11] G. F. Xue and X. C. Pang, A criterion for normal families of meromorphic functions, J. East China Norm. Univ. Natur. Sci. Ed. 2(1988), 15-22.
[12] L. Zalcman, Normal families:new perspectives, Bull.Amer.Math.Soc.35(1998), 215-230.
[13] L. Yang, Normality for families of meromorphic functions, Sci. Sinica, Ser. A 29(1986), 1263-1274.
[14] M. L. Fang and W. Hong, Some restults on normal family of meromorphic functions, Bull. Malays. Math. Sci. Soc. (2)23(2000), 143-151.
[15] J. B. Conway, Functions of one complex variable, Springer-Verlag, 1973.
[16] Schiwick. W, Normality criterion for families of meromorphic functions, J. Analyse Math. 52 (1989), 241-289.
[17] W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2)70(1959), 9-42.
[18] H.H. Chen and X. H. Hua, Normal families concerning shared values, Israel J. Math 115 (2000), 355-362.
[19] M. L. Fang and Y. Xu, Normal families of holomorphic functions and shared values, Israel J. Math, 2002, 129: 125-141.
[20] W. C. Lin and L. Z. Yang, Normality of a family of holomorphic functions which share one finite value with their derivatives, Acta Math. Sinica 2003, 46(3), 539-544.
[21] X. C. Pang, Shared values and normal families, Analysis 22(2002), 175-182.
[22] X. C. Pang and L. Zalcman, Normality and shared values, Ark. Mat., 38(2000),171-182.
[23] X. C. Pang and L. Zalcman, Normal families and shared values, The Bulletin of the London Mathematical Society 32(2000), 325-331.
[24] W. Schwick, Sharing values and normality, Archiv der Mathematik 59(1992), 50-54.
[25] Y. Xu, Normality criteria concerning sharing values, Indian Journal of Pure and Applied Mathematics 30(3) (1999), 287-293.
[2] J. Schiff, Normal Families, Springer-Verlag, 1993.
[3] L. Yang, Value Distribution Theory, Springer-Verlag&Science Press, Berlin, 1993.
[4] P. Montel, Sur les families de fonctions analytiques qui admettent des valuers exceptionnelles dans un domaine, Ann. cole Norm. Sup. (3)29(1912), 487-535.
[5] D. C. Sun, The shared value criterion for normality, J. Wuhan Univ. Natur. Sci. Ed. 3(1994), 9-12.
[6] Y. X. Gu, Un critère de normalité de fonctions méromorphes, Sci. Sinica, Special Issue 1(1979), 267-274.
[7] M. L. Fang and L. Zalcman, A note on normality and shared values, J. Austral Math. Soc, 76(2004), 141-150.
[8] L. Yang, A general criterion of normality, Acta Math. Sinica (New series) 1 (1985), 181-193.
[9] 朱经浩，亚纯函数的一个总的正规定则，科学通报，3(1986)，174-177。
[10] 王跃飞和方明亮，涉及零点重数的亚纯函数的值分布，数学学报，4(1998)，743-748．
[11] G. F. Xue and X. C. Pang, A criterion for normal families of meromorphic functions, J. East China Norm. Univ. Natur. Sci. Ed. 2(1988), 15-22.
[12] L. Zalcman, Normal families:new perspectives, Bull.Amer.Math.Soc.35(1998), 215-230.
[13] L. Yang, Normality for families of meromorphic functions, Sci. Sinica, Ser. A 29(1986), 1263-1274.
[14] M. L. Fang and W. Hong, Some restults on normal family of meromorphic functions, Bull. Malays. Math. Sci. Soc. (2)23(2000), 143-151.
[15] J. B. Conway, Functions of one complex variable, Springer-Verlag, 1973.
[16] Schiwick. W, Normality criterion for families of meromorphic functions, J. Analyse Math. 52 (1989), 241-289.
[17] W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2)70(1959), 9-42.
[18] H.H. Chen and X. H. Hua, Normal families concerning shared values, Israel J. Math 115 (2000), 355-362.
[19] M. L. Fang and Y. Xu, Normal families of holomorphic functions and shared values, Israel J. Math, 2002, 129: 125-141.
[20] W. C. Lin and L. Z. Yang, Normality of a family of holomorphic functions which share one finite value with their derivatives, Acta Math. Sinica 2003, 46(3), 539-544.
[21] X. C. Pang, Shared values and normal families, Analysis 22(2002), 175-182.
[22] X. C. Pang and L. Zalcman, Normality and shared values, Ark. Mat., 38(2000),171-182.
[23] X. C. Pang and L. Zalcman, Normal families and shared values, The Bulletin of the London Mathematical Society 32(2000), 325-331.
[24] W. Schwick, Sharing values and normality, Archiv der Mathematik 59(1992), 50-54.
[25] Y. Xu, Normality criteria concerning sharing values, Indian Journal of Pure and Applied Mathematics 30(3) (1999), 287-293.