Articles

UNIQUENESS OF MEROMORPHIC FUNCTIONS WHOSE NONLINEAR DIFFERENTIAL POLYNOMIALS SHARE ONE VALUE OR HAVE THE SAME FIXED POINTS IN AN ANGULAR DOMAIN

  • LI Xiao-Min ,
  • YI Hong-Xun
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  • Department of Mathematics, Ocean University of China, Qingdao 266100, China|Department of Physics and Mathematics, University of Eastern Finland P.O.Box 111, 80101 Joensuu, Finland; Department of Mathematics, Shandong University, Jinan 250100, China

Received date: 2009-08-24

  Revised date: 2013-04-10

  Online published: 2014-05-20

Supported by

The first author and the second author are supported by the NSFC (11171184) and the NSF of Shandong Province, China (Z2008A01).

Abstract

In this article, we study the uniqueness question of nonconstant meromorphic functions whose nonlinear differential polynomials share 1 or have the same fixed points in an angular domain. The results in this article improve Theorem 1 of Yang and Hua [26], and improve Theorem 1 of Fang and Qiu [6].

Cite this article

LI Xiao-Min , YI Hong-Xun . UNIQUENESS OF MEROMORPHIC FUNCTIONS WHOSE NONLINEAR DIFFERENTIAL POLYNOMIALS SHARE ONE VALUE OR HAVE THE SAME FIXED POINTS IN AN ANGULAR DOMAIN[J]. Acta mathematica scientia, Series B, 2014 , 34(3) : 593 -609 . DOI: 10.1016/S0252-9602(14)60032-1

References

[1] Bergweiler W, Eremenko A. On the singularities of the inverse to a meromorphic function of finite order. Rev Mat Iber, 1995, 11: 355–373

[2] Chuang C T. Sur les fonctions-types. Sci Sinica, 1961, 10: 171–181

[3] Chuang C T. On Borel directions of meromorphic functions of infinite order II. Bulletin of the Hongkong Mathematical Society, 1999, 2: 305–323

[4] Chen H H, Fang M L. On the value distribution of fnf′. Science in China, 1995, 38A(7): 789–798

[5] Fang M L. A note on a problem of Hayman. Analysis, 2000, 20: 45–49

[6] Fang M L, Qiu H L. Meromorphic functions that share fixed-points. J Math Anal Appl, 2000, 56: 426–439

[7] Gross F. On the distribution of values of meromorphic functions. Trans Amer Math Soc, 1968, 131: 199–214

[8] Gu Y X. Normal Families of Meromorphic Functions (Chinese). Sichuan: Sichuan Education Press, 1991

[9] Gundersen G G. Meromorphic functions that share three or four values. J London Math Soc, 1979, 20: 457–466

[10] Gundersen G G. Meromorphic functions that share four values. Trans Amer Math Soc, 1983, 227: 545–567
(Correction: 1987, 304: 847–850)

[11] Gundersen G G. Meromorphic functions that share three values IM and a fouth value CM. Complex Variables Theory Appl, 1992, 20: 99–106

[12] Gross F. Factorization of meromorphic functions and some probloms. Complex analysis (Proc Conf Univ Kentucky, Ky, 1976). Lecture Notes in Math, Vol 599, pp. 51–69. Berlin: Springer

[13] Gol’dberg A A, Ostrovskii I V. Value Distribution of Meromorphic Functions. 2nd Edition (Translations of
Mathematical Monographs). Vol 236. Providence, Rhode Island: American Mathematical Society, 2008

[14] Hayman W K. Meromorphic Functions. Oxford: The Clarendon Press, 1964

[15] Hayman W K. Picard values of meromorphic functions and their derivatives. Ann Math, 1959,70: 9–42

[16] Hiong K L. Sur les fonctions enti´eres et les fonctions (in Chinese). Beijing: Science Press, 1958

[17] Ishizaki K, Tohge K. On the complex oscillation of some linear differential equations. J Math Anal Appl, 1997, 206: 503–517

[18] Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin/New York: Walter de Gruyter, 1993

[19] Lin W C, Mori S, Tohge K. Uniqueness theorems in an angular domain. Tohoku Math J, 2006, 58: 509–527

[20] Lahiri I. Weighted value sharing and uniqueness of merommorphic functions. Complex Variables Theory Appl, 2001, 46: 241–253

[21] Mues E. Meromorphic functions sharing four values. Complex Variables Theory Appl, 1989, 12: 169–179

[22] Nevanlinna R. Le Th´eor`eme de Picard-Borel et la Th´eorie des Fonctions M´eromorphes. Paris: Gauthier-Villars, 1929

[23] Wu Z J, Sun D C. A remark on uniqueness theorems in an angular domain. Proc Japan Acad, 2008, 84A: 73–77

[24] Yang L. Value Distribution Theory. Berlin Heidelberg: Springer-Verlag, 1993

[25] Yi H X. On a question of Gross concerning uniqueness of entire functions. Bull Austral Math Soc, 1998, 57: 343–349

[26] Yang C C, Hua X H. Uniqueness and value-sharing of meromorphic functions. Ann Acad Sci Fenn Math, 1997, 22: 395–406

[27] Yang C C, Yi H X. Uniqueness Theory of Meromorphic Functions. Dordrecht/Boston/London: Kluwer Academic Publishers, 2003

[28] Yang C C. On deficiencies of differential polynomials II. Math Z, 1972, 125: 107–112

[29] Zheng J H. On uniqueness of meromorphic functions with shared values in one angular domains. Complex
Variables Theory Appl, 2003, 48: 777–785

[30] Zheng J H. On uniqueness of meromorphic functions with shared values in some angular domains. Canad J Math, 2004, 47: 152–160

[31] Zheng J H. Value Distribution of Meromorphic Functions. Beijing: Tsinghua University Press/Springer, 2010

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