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GROWTH OF MEROMORPHIC SOLUTIONS OF SOME ALGEBRAIC DIFFERENTIAL EQUATIONS

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  • School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China Department of Mathematics and Physics, Shanghai Dianji University, Shanghai 200240, China School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received date: 2014-02-11

  Online published: 2015-01-20

Supported by

The first author is supported by the NNSF of China (11101048). The second author is supported by the Tianyuan Youth Fund of the NNSF of China (11326083), the Shanghai University Young Teacher Training Program (ZZSDJ12020), the Innovation Program of Shanghai Municipal Education Commission (14YZ164) and and the Projects (13XKJC01) from the Leading Academic Discipline Project of Shanghai Dianji University. The third author is supported by the NNSF of China (11271090) and the NSF of Guangdong Province (S2012010010121).

Abstract

In this paper, by means of the normal family theory, we estimate the growth order of meromorphic solutions of some algebraic differential equations and improve the related result of Barsegian et al. [6]. We also give some examples to show that our results occur in some special cases.

Cite this article

LI Ye Zhou, QI Jian Ming, YUAN Wen Jun . GROWTH OF MEROMORPHIC SOLUTIONS OF SOME ALGEBRAIC DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia, Series B, 2015 , 35(1) : 105 -111 . DOI: 10.1016/S0252-9602(14)60143-0

References

[1] Bank S, Kaufman R. On meromorphic solutions of first-order differential equations. Comment Math Helv,
1976, 51: 289–299
[2] Barsegian G. Estimates of derivatives of meromorphic functions on sets of -points. J London Math Soc,
1986, 34: 534–540
[3] Barsegian G. On meromorphic solutions of algebraic differential equations. Complex Analysis and Math
Physics, Proceedings, Krasnoyarsk, 1987 (in Russian)
[4] Barsegian G. On a method of the study of algebraic differential equations of higher orders. Bull Hong Kong
Math Soc, 1998, 2(1): 159–164
[5] Barsegian G. Estimates of higher derivatives of meromorphic functions on sets of its a-points. Bull Hong
Kong Math Soc, 1999, 2(2): 341–346
[6] Barsegian G, Laine I, Yang C C. On a method of estimating derivatives in complex differential equations.
J Math Soc Japan, 2002, 54: 923–935
[7] Bergweiler W. On a theorem of Gol′dberg concerning meromorphic solutions of algebraic differential equations.
Complex Variables, 1998, 37: 93–96
[8] Clunie J, Hayman W K. The spherical derivative of integral and meromorphic functions. Comment Math
Helv, 1966, 40: 117–148
[9] Gol′dberg A A. On single-valued solutions of first-order differential equations (Russian). Ukra¨?n Mat Zh,
1956, (8): 254–261
[10] Gu R M, Li Z R, Yuan WJ. The growth of entire solutions of some algebraic differential equations. Georgian
Math J, 2011, 18(3): 489–495

[11] Gu R M, Ding J J, Yuan W J. On the estimate of growth order of solutions of a class of systems of algebraic
differential equations with higher orders. J Zhanjiang Normal Univ (in Chinese), 2009, 30(6): 39–43
[12] Frank G, Wang Y F. On the meromorphic solutions of algebraic differential equations. Analysis, 1998, 18:
49–54
[13] Hayman WK: Meromorphic Functions. Oxford: Clarendon Press, 1964
[14] He Y Z, Xiao X Z: Algebroid Functions and Ordinary Differential Equations (in Chinese). Beijing: Science
Press, 1988
[15] Laine I. Nevanlinna theory and complex differential equations. Berlin: de Gruyter, 1993
[16] Qi J M, Li Y Z, Yuan W J. Further results of Gol′dberg’s theorem concerning the growth of meromorphic
solutions of algebraic differential equations. Acta Math Sci, 2013, 33A(4): 759–765 (in Chinese)
[17] Yang L. Value Distribution Theory. Berlin: Springer-Verlag, 1993
[18] Yuan W J, Li Y Z, Lin J M. Growth of entire solutions of algebraic differential equations. Elec J Differ
Equ, 2012, 2012(94): 1–8
[19] Yuan W J, Xiao B, Zhang J J. The general result of Gol′dberg’s theorem concerning the growth of meromorphic
solutions of algebraic differential equations. Comput Appl Math, 2009, 58: 1788–1791
[20] Yuan W J, Li Z R, Zhang J J. On the estimate of growth of meromorphic solutions of some classes of higher
order algebraic differential equations. J Guangzhou Univ Nat Sci, 2009, 8(2): 28–31
[21] Zalcman L. A heuristic principle in complex function theory. Amer Math Monthly, 1975, 82: 813–817
[22] Zalcman L. Normal families: new perspectives. Bull Amer Math Soc, 1998, 35: 215-230

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