亚纯函数的差分多项式
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摘要
假设函数f(z)是亚纯函数,H(z,f)是关于f(z)的差分多项式,s(z)是关于f(z)的小函数,考察了差分多项式f(z)~nH(z,f)-s(z)的零点分布问题.首先得到了差分多项式f(z)~nH(z,f)-s(z)的零点计数函数和函数f(z)的特征函数以及极点计数函数之间的一些不等式估计,再根据这些不等式,建立了Hayman关于亚纯函数的一个经典结果的差分模拟.
In this paper, the authors investigate zeros of difference polynomials of the form f(z)~nH(z, f)-s(z), where f(z) is a meromorphic function, H(z,f) is a difference polynomial of f(z) and s(z) is a small function. The authors first obtain some inequalities for the relationship of the zero counting function of f(z)~nH(z,f)-s(z) and the characteristic function and pole counting function of f(z). Based on the above inequalities, the authors then establish some difference analogues of a classical result of Hayman for meromorphic functions.
In this paper, the authors investigate zeros of difference polynomials of the form f(z)~nH(z, f)-s(z), where f(z) is a meromorphic function, H(z,f) is a difference polynomial of f(z) and s(z) is a small function. The authors first obtain some inequalities for the relationship of the zero counting function of f(z)~nH(z,f)-s(z) and the characteristic function and pole counting function of f(z). Based on the above inequalities, the authors then establish some difference analogues of a classical result of Hayman for meromorphic functions.
引文
[1] Hayman W K. Meromorphic functions[M]. Oxford:Clarendon Press, 1964.
[2] Laine I. Nevanlinna theory and complex differential equations[M]. Berlin:W de Gruyter,1993.
[3] Hayman W K. Picard values of meromorphic functions and their derivatives[J]. Ann of Math, 1959, 70(2):9-42.
[4] 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(1):105-129.
[5] Halburd R G, Korhonen R J. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations[J].J Math Anal Appl, 2006,314(2):477-487.
[6] Halburd R G, Korhonen R J. Nevanlinna theory for the difference operator[J]. Ann Acad Sci Fenn Math, 2006, 31(2):463-478.
[7] Halburd R G,Korhonen R J,Tohge K. Holomorphic curves with shift-invariant hyperplane preimages[J].Trans Amer Math Soc,2014, 366(8):4267-4298.
[8] Laine I, Yang C C. Clunie theorems for difference and q-difference polynomials[J]. J Lond Math Soc, 2007, 76(3):556-566.
[9] Laine I,Yang C C. Value distribution of difference polynomials[J]. Proc Japan Acad,2007, 83(8):148-151.
[10] Liu K, Yang L Z. Value distribution of the difference operator[J]. Arch Math, 2009,92(3):270-278.
[11] Zheng X M, Chen Z X. On the value distribution of some difference polynomials[J]. J Math Anal Appl, 2013, 397(2):814-821.
[12] Li X M, Yi H X, Li W L. Value distribution of certain difference polynomials of meromorphic functions[J]. Rocky Mountain J Math, 2014, 44(2):599-632.
[13] Liu K, Liu X L, Cao T B. Value distributions and uniqueness of difference polynomials[J]. Adv Difference Equ, 2011, Article ID 234215, 12 pages.
[14] Ablowitz M J, Halburd R G, Herbst B. On the extension of the Painleve property to difference equations[J]. Nonlinearity, 2000, 13(3):889-905.
[15] Gol'dberg A A, Ostrovskii I V. Distribution of values of meromorphic functions[M].Moscow:Nauka, 1970.
[16]张然然,陈宗煊.亚纯函数养分多项式的值分布[J].中国科学:数学,2012,42(11):1115-1130.
[17] Zheng X M, Chen Z X. Some properties of meromorphic solutions of q-difference equations[J]. J Math Anal Appl, 2010, 361(2):472-480.
[18] Li P, Wang W J. Entire functions that share a small function with its derivative[J]. J Math Anal Appl, 2007, 328(1):743-751.
[2] Laine I. Nevanlinna theory and complex differential equations[M]. Berlin:W de Gruyter,1993.
[3] Hayman W K. Picard values of meromorphic functions and their derivatives[J]. Ann of Math, 1959, 70(2):9-42.
[4] 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(1):105-129.
[5] Halburd R G, Korhonen R J. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations[J].J Math Anal Appl, 2006,314(2):477-487.
[6] Halburd R G, Korhonen R J. Nevanlinna theory for the difference operator[J]. Ann Acad Sci Fenn Math, 2006, 31(2):463-478.
[7] Halburd R G,Korhonen R J,Tohge K. Holomorphic curves with shift-invariant hyperplane preimages[J].Trans Amer Math Soc,2014, 366(8):4267-4298.
[8] Laine I, Yang C C. Clunie theorems for difference and q-difference polynomials[J]. J Lond Math Soc, 2007, 76(3):556-566.
[9] Laine I,Yang C C. Value distribution of difference polynomials[J]. Proc Japan Acad,2007, 83(8):148-151.
[10] Liu K, Yang L Z. Value distribution of the difference operator[J]. Arch Math, 2009,92(3):270-278.
[11] Zheng X M, Chen Z X. On the value distribution of some difference polynomials[J]. J Math Anal Appl, 2013, 397(2):814-821.
[12] Li X M, Yi H X, Li W L. Value distribution of certain difference polynomials of meromorphic functions[J]. Rocky Mountain J Math, 2014, 44(2):599-632.
[13] Liu K, Liu X L, Cao T B. Value distributions and uniqueness of difference polynomials[J]. Adv Difference Equ, 2011, Article ID 234215, 12 pages.
[14] Ablowitz M J, Halburd R G, Herbst B. On the extension of the Painleve property to difference equations[J]. Nonlinearity, 2000, 13(3):889-905.
[15] Gol'dberg A A, Ostrovskii I V. Distribution of values of meromorphic functions[M].Moscow:Nauka, 1970.
[16]张然然,陈宗煊.亚纯函数养分多项式的值分布[J].中国科学:数学,2012,42(11):1115-1130.
[17] Zheng X M, Chen Z X. Some properties of meromorphic solutions of q-difference equations[J]. J Math Anal Appl, 2010, 361(2):472-480.
[18] Li P, Wang W J. Entire functions that share a small function with its derivative[J]. J Math Anal Appl, 2007, 328(1):743-751.