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ON THE TRANSCENDENTAL ENTIRE SOLUTIONS OF A CLASS OF DIFFERENTIAL EQUATIONS

  • Lu, Weiran (Department of Mathematics China University of Petroleum) ;
  • Li, Qiuying (Department of Mathematics China University of Petroleum) ;
  • Yang, Chungchun (Department of Mathematics China University of Petroleum)
  • Received : 2011.11.09
  • Published : 2014.09.30

Abstract

In this paper, we consider the differential equation $$F^{\prime}-Q_1=Re^{\alpha}(F-Q_2)$$, where $Q_1$ and $Q_2$ are polynomials with $Q_1Q_2{\neq}0$, R is a rational function and ${\alpha}$ is an entire function. We consider solutions of the form $F=f^n$, where f is an entire function and $n{\geq}2$ is an integer, and we prove that if f is a transcendental entire function, then $\frac{Q_1}{Q_2}$ is a polynomial and $f^{\prime}=\frac{Q_1}{nQ_2}f$. This theorem improves some known results and answers an open question raised in [16].

References

  1. R. Bruck, On entire functions which share one value CM with their first derivatives, Results Math. 30 (1996), no. 1-2, 21-24. https://doi.org/10.1007/BF03322176
  2. Z. X. Chen and K. H. Shon, On conjecture of R. Bruck concerning the entire function sharing one value CM with its derivative, Taiwanese J. Math. 8 (2004), no. 2, 235-244. https://doi.org/10.11650/twjm/1500407625
  3. J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17-22.
  4. W. Doeringer, Exceptional value of differential polynomials, Pacific J. Math. 98 (1982), no. 1, 55-52. https://doi.org/10.2140/pjm.1982.98.55
  5. G. G. Gundersen and L. Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998), no. 1, 88-95. https://doi.org/10.1006/jmaa.1998.5959
  6. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
  7. I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
  8. C. L. Lei, M. L. Fang, et al., A note on unicity of meromorophic functions, Acta Math. Sci. Ser. A. Chin. Ed. 28 (2008), no. 5, 802-807.
  9. F. Lu, A note on the Bruck conjecture, Bull. Korean Math. Soc. 48 (2011), no. 5, 951-957. https://doi.org/10.4134/BKMS.2011.48.5.951
  10. F. Lu and H. X. Yi, The Bruck conjecture and entire functions sharing polynomals with their k−th derivatives, J. Korean Math. Soc. 48 (2011), no. 3, 499-512. https://doi.org/10.4134/JKMS.2011.48.3.499
  11. E. Mues and N. Steinmetz, Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen, Manuscripta Math. 29 (1979), no. 2-4, 195-206. https://doi.org/10.1007/BF01303627
  12. L. A. Rubel and C. C. Yang, Values shared by an entire function and its derivative, Complex Analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), 101-103, Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977.
  13. L. Z. Yang and J. L. Zhang, Non-existence of meromorphic solutions of a Fermat type functional equation, Aequationes Math. 76 (2008), no. 1-2, 140-150. https://doi.org/10.1007/s00010-007-2913-7
  14. H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, 1995.
  15. J. L. Zhang and L. Z. Yang, A power of an entire function sharing one value with its derivative, Comput. Math. Appl. 60 (2010), no. 7, 2153-2160. https://doi.org/10.1016/j.camwa.2010.08.001
  16. J. L. Zhang and L. Z. Yang, A power of a meromorphic function sharing a small function with its derivative, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 1, 249-260.
  17. T. D. Zhang and W. R. Lu, Notes on a meromorphic function sharing one small function with its derivative, Complex Var. Elliptic Equ. 53 (2008), no. 9, 857-867. https://doi.org/10.1080/17476930802166386

Cited by

  1. A RESULT ON A CONJECTURE OF W. LÜ, Q. LI AND C. YANG vol.53, pp.2, 2016, https://doi.org/10.4134/BKMS.2016.53.2.411