Journal of East China Normal University(Natural Science) >

2015 , Vol. 2015 >Issue 3: 1 - 8

DOI: https://doi.org/10.3969/j.issn1000-5641.2015.03.001

Article

On the value distribution of complex difference

  • ZHOU Li-Li ,
  • HUANG Zhi-Gang ,
  • SUN Gui-Rong
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Received date: 2014-10-20

  Online published: 2015-05-28

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Abstract

In this paper, we investigate the value distribution of difference $\varphi_1 (z)=\frac{f(z+c)}{(f(z))^k}-a(f(z))^n$ and \varphi _2 (z)=\frac{\prod\limits_{i=1}^m {f(z+c_i )}}{(f(z))^k}-a(f(z))^n. From the viewpoint of the Nevanlinna theory, we obtained some results in analogy with Picard Theorem.

Key words： complexdifference; difference polynomial; transcendentalmeromorphic function; the growth of order

Cite this article

ZHOU Li-Li , HUANG Zhi-Gang , SUN Gui-Rong . On the value distribution of complex difference[J]. Journal of East China Normal University(Natural Science), 2015 , 2015(3) : 1 -8 . DOI: 10.3969/j.issn1000-5641.2015.03.001

References



[1]HAYMAN W K. Meromorphic Functions [M]. Oxford: Clarendon Press,1964.
[2]LAINE I. Nevanlinna Theory and Complex Differential Equations [M].Berlin: Walter de Gruyter, 1993.
[3]YANG L. Value Distribution Theory [M]. Berlin: Springer-Verlag,Science Press, 1993.
[4]HALBURD R G, KORHONEN R J. Meromorphic solutions of differenceequations, Integrability and the discrete Painleve equations [J]. JPhys, 2007, A40. 1-38.
[5]HEITTOKANGAS J, KORHONEN R, LAINE I, et al. Complex differenceequations of Malmquist type [J]. Comput Methods Funct Theory, 2001,1(1): 27-39.
[6]LAINE I, YANG C C. Value distribution of difference polynomials [J].Proc Japan Acad Ser 2007, A83: 148-151.
[7]ZHENG X M, CHEN Z X. On the value distribution of Some difference polynomials [J]. J Math Anal Appl, 2013, 397: 814-821.
[8]CHIANG Y M, FENG S J. On the Nevanlinna characteristic of f(z+c)and difference quotients in the complex plane [J]. Ramanujan J,2008, 16: 105-129.
[9]HALBURD R G, KORHONEN R J. Difference analogue of the lemma on thelogarithmic derivative with applications to difference equations[J]. J Math Anal Appl, 2006, 314: 477-487.
[10]张然然, 陈宗煊. 亚纯函数差分多项式的值分布~[J]. 中国科学: 数学,2012, 42(11): 1115-1130.
[11]ABILWITE M J, HALBURD R G, HERST B. On the extension of the Painleveproperty to difference equations [J]. Nonlinearity, 2000, 13:889-905.
[12]BERGWEILER W, LANGLEY J K. Zeros of differenxes of meromorphicfunctions [J]. Math Proc Cambridge Philos Soc, 2007, 142: 133-147.
[13]CHEN Z X, HUANG Z B, ZHENG X M. On properties of differencepolynomials [J]. Acta Math Sci, 2011, 31B(2): 627-633.
[14]CHEN Z X, SHON K H. Estimates for the zeros of differences ofmeromorphic functions [J]. Sci China, 2009, 52A(11): 2447-2458.
[15]CHEN Z X, SHON K H. On zeros and fixed points of differences ofmeromorphic functions [J]. J Math Anal Appl, 2008, 344(1): 373-383.
[16]CHIANG Y M, FENG S J. On the growth of logarithmic differences,difference quotients and logarithmic derivatives of meromorphicfunctions [J]. Trans Amer Math Soc, 2009, R361(7): 3767-3791.
[17]HALBURD R G, KORHONEN R J. Existence of finite-order meromorphicsolutions as a detector of integrability in difference equations[J]. Physica, 2006, 218D: 191-203.
[18]ISHIZAKI K, YANAGIHARA N. Wiman-Valiron method for differenceequations [J]. Nagoya Math J, 2004, 175: 75-102.
[19]LIU K. Meromorphic functions sharing a set with applications todifference equations [J]. J Math Anal Appl, 2009, 359: 384-393.
[20]ZHENG X M, CHEN Z X. Growth of meromorphic solutions of lineardifference equations [J]. J Math Appl, 2011, 384(2): 349-356.
[21]ZHENG X M, CHEN Z X. Some properties of meromorphic solutions of$q$-difference equations [J]. J Math Appl, 2010, 361(2): 472-480.
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