涉及差分多项式的值分布和唯一性问题
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摘要
本文在条件N(r,1/f)+N(r,f)=S(r,f)下研究了超级小于1的亚纯函数f的1类非线性差分多项式的值分布问题,改进了Laine-Yang~([4])中的相应结果.本文也分别研究了具有1个非零公共值的1类非线性微分多项式和差分多项式的唯一性问题,推广了Yang-Hua~([14])和方明亮~([15])中的相应结果。
Under the assumption N(r,1/f)+N(r,f)=S(r,f),we study a value distribution question of certain nonlinear difference polynomials of meromorphic functions of hyper-order less than one.The results in this paper improve the corresponding results given by Laine-Yang~([4]).We also study a uniqueness question of meromorphic functions whose certain nonlinear differential polynomials or certain difference polynomials share a finite nonzero value.Also the uniqueness results of this paper improve the corresponding results given by Yang-Hua~([14]) and Fang~([15]).
Under the assumption N(r,1/f)+N(r,f)=S(r,f),we study a value distribution question of certain nonlinear difference polynomials of meromorphic functions of hyper-order less than one.The results in this paper improve the corresponding results given by Laine-Yang~([4]).We also study a uniqueness question of meromorphic functions whose certain nonlinear differential polynomials or certain difference polynomials share a finite nonzero value.Also the uniqueness results of this paper improve the corresponding results given by Yang-Hua~([14]) and Fang~([15]).
引文
[1] Hayman W K.Meromorphic Functions[M].Oxford:The Claredon,1964.
[2]仪洪勋,杨重骏.亚纯函数的唯一性理论[M].北京:科学出版社,1995.Yi H X,Yang C C.Uniqueness Theory of Meromorphic Functions[M].Beijing:Science Press,1995.
[3] Yi H X.On a Theorem of Tumura and Clunie for a Differential Polynomial[J].Bull London Math Soc,1988,20(6):593-596.
[4] Laine I,Yang C C.Value distribution of difference polynomials[J].Proc Japan Acad Ser A Math Sci,2007,83(8):148-151.
[5] Lahiri I.Weighted sharing of three values and uniqueness of meromorphic functions[J].Kodai Math J,2001,24(3):421-435.
[6] Hayman W K.Picard values of meromorphic functions and their derivatives[J].Ann Math,1959,70(2):9-42.
[7] Yang C C.On deficiencies of differential polynomials II[J].Math Z,1972,125(2):107-112.
[8]Halburd R,Korhonen R,Tohge K.Holomorphic curves with shift-invariant hyperplane preimages[J].Trans Amer Math Soc,2014,366:4267-4298.
[9] Chiang Y M,Feng S J.On the Nevanlinna characteristic of f(z+η)and difference equations in the complex plane[J].Ramanujan J,2008,16:105-129.
[10] Laine I,Yang C C.Clunie theorems for difference and q-difference polynomials[J].J London Math Soc,2007,76:556-566.
[11] Heittokangas J,Korhonen R,Laine I,et al.Value sharing results for shifts of meromorphic functions and sufficient conditions for periodicity[J].J Math Anal Appl,2009,355:352-363.
[12] Laine I,Yang C C.Value distribution of difference polynomials[J].Proc Japan Acad Ser A Math Sci,2007,83(8):148-151.
[13] Lahiri I.Uniqueness of meromorphic functions as governed by their differential polynomials[J].Yokohama Math J,1997,44:147-156.
[14] Yang C C,Hua X H.Uniqueness and value-sharing of meromorphic functions[J].Ann Acad Sci Fenn Math,1997,22:95-406.
[15] Fang M L.Uniqueness and value sharing of entire functions[J].Comput Math Appl,2002,44:823-831.
[16] Mues E,Steinmetz N.The theorem of Tumura-Clunie for meromorphic functions[J].J London Math Soc,1981,23(2):113-122.
[2]仪洪勋,杨重骏.亚纯函数的唯一性理论[M].北京:科学出版社,1995.Yi H X,Yang C C.Uniqueness Theory of Meromorphic Functions[M].Beijing:Science Press,1995.
[3] Yi H X.On a Theorem of Tumura and Clunie for a Differential Polynomial[J].Bull London Math Soc,1988,20(6):593-596.
[4] Laine I,Yang C C.Value distribution of difference polynomials[J].Proc Japan Acad Ser A Math Sci,2007,83(8):148-151.
[5] Lahiri I.Weighted sharing of three values and uniqueness of meromorphic functions[J].Kodai Math J,2001,24(3):421-435.
[6] Hayman W K.Picard values of meromorphic functions and their derivatives[J].Ann Math,1959,70(2):9-42.
[7] Yang C C.On deficiencies of differential polynomials II[J].Math Z,1972,125(2):107-112.
[8]Halburd R,Korhonen R,Tohge K.Holomorphic curves with shift-invariant hyperplane preimages[J].Trans Amer Math Soc,2014,366:4267-4298.
[9] Chiang Y M,Feng S J.On the Nevanlinna characteristic of f(z+η)and difference equations in the complex plane[J].Ramanujan J,2008,16:105-129.
[10] Laine I,Yang C C.Clunie theorems for difference and q-difference polynomials[J].J London Math Soc,2007,76:556-566.
[11] Heittokangas J,Korhonen R,Laine I,et al.Value sharing results for shifts of meromorphic functions and sufficient conditions for periodicity[J].J Math Anal Appl,2009,355:352-363.
[12] Laine I,Yang C C.Value distribution of difference polynomials[J].Proc Japan Acad Ser A Math Sci,2007,83(8):148-151.
[13] Lahiri I.Uniqueness of meromorphic functions as governed by their differential polynomials[J].Yokohama Math J,1997,44:147-156.
[14] Yang C C,Hua X H.Uniqueness and value-sharing of meromorphic functions[J].Ann Acad Sci Fenn Math,1997,22:95-406.
[15] Fang M L.Uniqueness and value sharing of entire functions[J].Comput Math Appl,2002,44:823-831.
[16] Mues E,Steinmetz N.The theorem of Tumura-Clunie for meromorphic functions[J].J London Math Soc,1981,23(2):113-122.