DOI QR코드

DOI QR Code

MEROMORPHIC FUNCTIONS SHARING A NONZERO POLYNOMIAL CM

  • Li, Xiao-Min (DEPARTMENT OF MATHEMATICS OCEAN UNIVERSITY OF CHINA) ;
  • Gao, Ling (DEPARTMENT OF MATHEMATICS OCEAN UNIVERSITY OF CHINA)
  • Published : 2010.03.31

Abstract

In this paper, we prove that if $f^nf'\;-\;P$ and $g^ng'\;-\;P$ share 0 CM, where f and g are two distinct transcendental meromorphic functions, $n\;{\geq}\;11$ is a positive integer, and P is a nonzero polynomial such that its degree ${\gamma}p\;{\leq}\;11$, then either $f\;=\;c_1e^{cQ}$ and $g\;=\;c_2e^{-cQ}$, where $c_1$, $c_2$ and c are three nonzero complex numbers satisfying $(c_1c_2)^{n+1}c^2\;=\;-1$, Q is a polynomial such that $Q\;=\;\int_o^z\;P(\eta)d{\eta}$, or f = tg for a complex number t such that $t^{n+1}\;=\;1$. The results in this paper improve those given by M. L. Fang and H. L. Qiu, C. C. Yang and X. H. Hua, and other authors.

Keywords

References

  1. T. C. Alzahary and H. X. Yi, Weighted sharing three values and uniqueness of meromorphic functions, J. Math. Anal. Appl. 295 (2004), no. 1, 247–257. https://doi.org/10.1016/j.jmaa.2004.03.040
  2. W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), no. 2, 355–373.
  3. W. Bergweiler and X. C. Pang, On the derivative of meromorphic functions with multiple zeros, J. Math. Anal. Appl. 278 (2003), no. 2, 285–292. https://doi.org/10.1016/S0022-247X(02)00349-9
  4. H. H. Chen and M. L. Fang, The value distribution of $f^n$ f', Sci. China Ser. A 38 (1995), no. 7, 789–798.
  5. M. L. Fang, A note on a problem of Hayman, Analysis (Munich) 20 (2000), no. 1, 45–49.
  6. M. L. Fang and H. L. Qiu, Meromorphic functions that share fixed-points, J. Math. Anal. Appl. 268 (2002), no. 2, 426–439. https://doi.org/10.1006/jmaa.2000.7270
  7. W. K. Hayman, Meromorphic Functions, The Clarendon Press,Oxford, 1964.
  8. W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2) 70 (1959), 9–42. https://doi.org/10.2307/1969890
  9. I. Lahiri and A. Sarkar, Uniqueness of a meromorphic function and its derivative, JIPAM. J. Inequal. Pure Appl. Math. 5 (2004), no. 1, Article 20, 9 pp.
  10. I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
  11. A. Z. Mokhonko, On the Nevanlinna characteristics of some meromorphic functions, Theory of Functions, Functional Analysis and Their Applications, vol.14, Izd-vo Khar’kovsk. Un-ta, 1971, pp. 83–87.
  12. E. Mues, Meromorphic functions sharing four values, Complex Variables Theory Appl. 12 (1989), no. 1-4, 169–179. https://doi.org/10.1080/17476938908814363
  13. R. Nevanlinna, Einige Eindeutigkeitssatze in der Theorie der Meromorphen Funktionen, Acta Math. 48 (1926), no. 3-4, 367–391. https://doi.org/10.1007/BF02565342
  14. C. C. Yang, On deficiencies of differential polynomials. II, Math. Z. 125 (1972), 107– 112. https://doi.org/10.1007/BF01110921
  15. C. C. Yang and X. H. Hua, Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (1997), no. 2, 395–406.
  16. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, Dordrecht/Boston/London, 2003.
  17. L. Yang, Value Distribution Theory, Springer-Verlag, Berlin Heidelberg, 1993.
  18. L. Yang, Normality for families of meromorphic functions, Sci. Sinica Ser. A 29 (1986), no. 12, 1263–1274.
  19. L. Zalcman, On some problems of Hayman, Preprint (Bar-Ilan University), 1995.
  20. Q. C. Zhang, Meromorphic functions sharing three values, Indian J. Pure Appl. Math. 30 (1999), no. 7, 667–682.

Cited by

  1. Uniqueness of meromorphic functions sharing a nonzero polynomial with finite weight vol.34, pp.1, 2013, https://doi.org/10.1134/S1995080213010101
  2. Uniqueness of meromorphic functions sharing two values vol.2012, pp.1, 2012, https://doi.org/10.1186/1029-242X-2012-100
  3. Uniqueness of meromorphic functions sharing one value or fixed points vol.49, pp.6, 2014, https://doi.org/10.3103/S1068362314060119
  4. Meromorphic Functions Sharing a Nonzero Polynomial IM vol.53, pp.2, 2013, https://doi.org/10.5666/KMJ.2013.53.2.191