Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A RESULT ON A CONJECTURE OF W. LÜ, Q. LI AND C. YANG

  • Received : 2014.10.23
  • Published : 2016.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate the problem of transcendental entire functions that share two values with one of their derivative. Let f be a transcendental entire function, n and k be two positive integers. If $f^n-Q_1$ and $(f^n)^{(k)}-Q_2$ share 0 CM, and $n{\geq}k+1$, then $(f^n)^{(k)}{\equiv}{\frac{Q_2}{Q_1}}f^n$. Furthermore, if $Q_1=Q_2$, then $f=ce^{\frac{\lambda}{n}z}$, where $Q_1$, $Q_2$ are polynomials with $Q_1Q_2{\not\equiv}0$, and c, ${\lambda}$ are non-zero constants such that ${\lambda}^k=1$. This result shows that the Conjecture given by W. $L{\ddot{u}}$, Q. Li and C. Yang [On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281-1289.] is true. Also we exhibit some examples to show that the conditions of our result are the best possible.

Keywords

References

  1. R. Bruck, On entire functions which share one value CM with their first derivative, Results Math. 30 (1996), no. 1-2, 21-24. https://doi.org/10.1007/BF03322176
  2. J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17-22.
  3. G. G. Gundersen, Meromorphic functions that share finite values with their derivative, J. Math. Anal. Appl. 75 (1980), no. 2, 441-446. https://doi.org/10.1016/0022-247X(80)90092-X
  4. 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
  5. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
  6. F. Lu and H. X. Yi, The Bruck conjecture and entire functions sharing polynomials 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
  7. W. Lu, Q. Li, and C. Yang, On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281-1289. https://doi.org/10.4134/BKMS.2014.51.5.1281
  8. E. Mues and N. Steinmetz, Meromorphe Funktionen, die mit ihrer ersten und zweiten Ableitung einen endlichen Wert teilen, Complex Var. Theory Appl. 6 (1986), no. 1, 51-71. https://doi.org/10.1080/17476938608814158
  9. 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), pp. 101-103. Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977.
  10. C. C. Yang, On deficiencies of differential polynomials. II, Math. Z. 125 (1972), 107-112. https://doi.org/10.1007/BF01110921
  11. L. Z. Yang, Entire functions that share finite values with their derivatives, Bull. Austral. Math. Soc. 41 (1990), no. 3, 337-342. https://doi.org/10.1017/S0004972700018190
  12. 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
  13. H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, 1995.
  14. J. L. Zhang, Meromorphic functions sharing a small function with their derivatives, Kyungpook Math. J. 49 (2009), no. 1, 143-154. https://doi.org/10.5666/KMJ.2009.49.1.143
  15. 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.
  16. 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