基于分担值的亚纯函数唯一性与正规族
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摘要
本文主要运用Nevanlinna值分布理论和Zalcman引理分别就亚纯函数的唯一性和正规族问题进行了探讨和研究。
在唯一性方面,利用加权的思想研究了整函数及其微分多项式分担一个值的唯一性问题。所得结果完善和推广了现有的一些结论。在正规族方面,主要探讨了分担一函数的亚纯函数族的正规性,得到了相应的正规定则,推广了当前的结果。
In this thesis, Mainly by using the Nevanlinna’value distribution theory and Zalcman’s lemma, the problems of uniqueness of meromorphic functions and the normal families are studied.
With regard to uniqueness, this dissertation studies the problem of entire functions and their differential polynomial sharing one value by using the idea of weighted sharing. The results obtained in this thesis perfect and generalize some of the present conclusions. Concerning normal families of meromorphic functions, normal families with sharing values has mainly discussed. And some results are obtained which generalize and improve the present result.
在唯一性方面,利用加权的思想研究了整函数及其微分多项式分担一个值的唯一性问题。所得结果完善和推广了现有的一些结论。在正规族方面,主要探讨了分担一函数的亚纯函数族的正规性,得到了相应的正规定则,推广了当前的结果。
In this thesis, Mainly by using the Nevanlinna’value distribution theory and Zalcman’s lemma, the problems of uniqueness of meromorphic functions and the normal families are studied.
With regard to uniqueness, this dissertation studies the problem of entire functions and their differential polynomial sharing one value by using the idea of weighted sharing. The results obtained in this thesis perfect and generalize some of the present conclusions. Concerning normal families of meromorphic functions, normal families with sharing values has mainly discussed. And some results are obtained which generalize and improve the present result.
引文
[1]仪洪勋,杨重骏.亚纯函数唯一性理论[M].北京:科学出版社,2006(1995).
[2]顾永兴,庞学诚,方明亮.正规族理论及应用[M].北京:科学出版社, 2007.
[3]顾永兴,亚纯函数的正规性[M].四川:四川教育出版社,1991.
[4] Q.L.Xiong.Sur les fonctions holomorphes dont les derivees admettent une valeur exceptionnelle [J]. Ann.Ecole Norm.Sup.,1955,72:165-197.
[5]李江涛,顾永兴.亚纯函数及其导数的唯一性[J].数学学报,2000,Vol.43(1).No:1.
[6] H.Milloux.Extension d’un theoreme de M.R.Nevanlinna et applications[J]. Act.Scient.et Ind.,1940:888.
[7] W.K. Hayman. Picard Values of meromorphic functions and their derivatives[J]. Ann. of Math., 1959,70:9-42.
[8]方企勤.复变函数教程[M].北京:北京大学出版社,1996.
[9]张广厚.整函数和亚纯函数理论[M].北京:科学出版社,1986.
[10] R.Nevanlinna.Le Theoreme de Picard-Borel et la theorie des fonctions meromorphes[M]. Paris,1992.
[11]李松鹰,谢晖春.关于亚纯函数的正规族[J].数学学报, 1986,29:468-477.
[12] Y.X.Gu.A normal criterion of meromorphic families[J].Sci.Sinica, 1979,(I):267-274.
[13] Y.Xu.Normality criteria concerning sharing values[J].Indian Pure Appl.Math,1999,30: 287-293.
[14] Zalcman.L. A heuristic principle in comple function theory [J]. Amer. Math.Monthly,1975,82: 813-875.
[15]庞学诚. Bloch原则和正规定则[J].中国科学(A辑), 1989, 32:782-791.
[16] W.Schwick,Normality Crietion for families of meromorphic functions[J].Analyses Math, 1989,52:241-289.
[17]陈怀惠,顾永兴,Marty定则的改进及应用[J].中国科学(A辑),1993, 23:123-129.
[18] X.C.Pang, L.Zalcman.Normal families and shared values[J].Bull.London Math.Soc.,2000, 32 325-331.
[19] I.Lahiri.Weighted Sharing and Uniqueness of Meromorphic Functions[J].Nagoya Math.J., 2001,161:193-206.
[20] M.L.Fang.Uniqueness and Value-Sharing of Entire Functions[J].Comput.Math.Appl.,2002,44.
[21] L.Zalcman.Normal families.New perspectives[M].Bull.Amer.Math.Soc.,1998,35:215-230.
[22] J.Li,Q.Lu,Uniqueness of meromorphic functions and differential polynomials sharing onevalue IM[J].Math.Anal.Appl.,2002, 44:607-617.
[23] S.S.Bhoosnurmath,R.S.Dyavanal,Uniqueness and value of meromorphic functions[J]. Comput.Math.Appl.,2007,53:1191-1205.
[24] X.Y.Zhang.Entire or mormorphic functions share one value[J].Compu.Math.Appl.,2008,56.
[25] C.C.Yang,On deficiencies of differential polynomials II[J].Math.Z.,1972,149:107-112.
[26] Q.C.Zheng, Meromorphic function that share one small function with its derivate[J].Inequal. Pure.Appl.Math,2005,6(4).
[27] G.Frank.Entire Vermutung Hayman uber nullstellen meromorpher Function[J]. math.Z.,1976, 194: 29-36.
[28] W.Schwick,Exceptional functions and normality[J],Bull.London Math.Soc.,1997,29:425-432.
[29] Y.F.Wang,M.L.Fang,Picard values and normal families of meromorphic functions with multiple zeros[J].Acta Math.Sinica(Chin.Ser),1998,41:743-748.
[30] M.L.Fang,L.Zalcman.A note on normality and shared values[J].Aust.Math.Soc.,2004,76: 141-150.
[31] Q.Zhang,Normal families of meromorphic functions concerning shared values [J].Math.Anal. Appl.,2008,338:545-551.
[32] Y.T.Li,Y.X.Gu.On normal families of meromorphic functions[J].Math.Anal.Appl.,2009,354: 421-425.
[2]顾永兴,庞学诚,方明亮.正规族理论及应用[M].北京:科学出版社, 2007.
[3]顾永兴,亚纯函数的正规性[M].四川:四川教育出版社,1991.
[4] Q.L.Xiong.Sur les fonctions holomorphes dont les derivees admettent une valeur exceptionnelle [J]. Ann.Ecole Norm.Sup.,1955,72:165-197.
[5]李江涛,顾永兴.亚纯函数及其导数的唯一性[J].数学学报,2000,Vol.43(1).No:1.
[6] H.Milloux.Extension d’un theoreme de M.R.Nevanlinna et applications[J]. Act.Scient.et Ind.,1940:888.
[7] W.K. Hayman. Picard Values of meromorphic functions and their derivatives[J]. Ann. of Math., 1959,70:9-42.
[8]方企勤.复变函数教程[M].北京:北京大学出版社,1996.
[9]张广厚.整函数和亚纯函数理论[M].北京:科学出版社,1986.
[10] R.Nevanlinna.Le Theoreme de Picard-Borel et la theorie des fonctions meromorphes[M]. Paris,1992.
[11]李松鹰,谢晖春.关于亚纯函数的正规族[J].数学学报, 1986,29:468-477.
[12] Y.X.Gu.A normal criterion of meromorphic families[J].Sci.Sinica, 1979,(I):267-274.
[13] Y.Xu.Normality criteria concerning sharing values[J].Indian Pure Appl.Math,1999,30: 287-293.
[14] Zalcman.L. A heuristic principle in comple function theory [J]. Amer. Math.Monthly,1975,82: 813-875.
[15]庞学诚. Bloch原则和正规定则[J].中国科学(A辑), 1989, 32:782-791.
[16] W.Schwick,Normality Crietion for families of meromorphic functions[J].Analyses Math, 1989,52:241-289.
[17]陈怀惠,顾永兴,Marty定则的改进及应用[J].中国科学(A辑),1993, 23:123-129.
[18] X.C.Pang, L.Zalcman.Normal families and shared values[J].Bull.London Math.Soc.,2000, 32 325-331.
[19] I.Lahiri.Weighted Sharing and Uniqueness of Meromorphic Functions[J].Nagoya Math.J., 2001,161:193-206.
[20] M.L.Fang.Uniqueness and Value-Sharing of Entire Functions[J].Comput.Math.Appl.,2002,44.
[21] L.Zalcman.Normal families.New perspectives[M].Bull.Amer.Math.Soc.,1998,35:215-230.
[22] J.Li,Q.Lu,Uniqueness of meromorphic functions and differential polynomials sharing onevalue IM[J].Math.Anal.Appl.,2002, 44:607-617.
[23] S.S.Bhoosnurmath,R.S.Dyavanal,Uniqueness and value of meromorphic functions[J]. Comput.Math.Appl.,2007,53:1191-1205.
[24] X.Y.Zhang.Entire or mormorphic functions share one value[J].Compu.Math.Appl.,2008,56.
[25] C.C.Yang,On deficiencies of differential polynomials II[J].Math.Z.,1972,149:107-112.
[26] Q.C.Zheng, Meromorphic function that share one small function with its derivate[J].Inequal. Pure.Appl.Math,2005,6(4).
[27] G.Frank.Entire Vermutung Hayman uber nullstellen meromorpher Function[J]. math.Z.,1976, 194: 29-36.
[28] W.Schwick,Exceptional functions and normality[J],Bull.London Math.Soc.,1997,29:425-432.
[29] Y.F.Wang,M.L.Fang,Picard values and normal families of meromorphic functions with multiple zeros[J].Acta Math.Sinica(Chin.Ser),1998,41:743-748.
[30] M.L.Fang,L.Zalcman.A note on normality and shared values[J].Aust.Math.Soc.,2004,76: 141-150.
[31] Q.Zhang,Normal families of meromorphic functions concerning shared values [J].Math.Anal. Appl.,2008,338:545-551.
[32] Y.T.Li,Y.X.Gu.On normal families of meromorphic functions[J].Math.Anal.Appl.,2009,354: 421-425.