Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SETS AND VALUE SHARING OF q-DIFFERENCES OF MEROMORPHIC FUNCTIONS

  • Received : 2012.02.05
  • Published : 2013.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate uniqueness problems of certain types of $q$-difference polynomials, which improve some results in [20]. However, our proof is different from that in [20]. Moreover, we obtain a uniqueness result in the case where $q$-differences of two entire functions share values as well. This research also shows that there exist two sets, such that for a zero-order non-constant meromorphic function $f$ and a non-zero complex constant $q$, $E(S_j,f)=E(S_j,{\Delta}_qf)$ for $j=1,2$ imply $f(z)=t{\Delta}_qf$, where $t^n=1$. This gives a partial answer to a question of Gross concerning a zero order meromorphic function $f(z)$ and $t{\Delta}_qf$.

Keywords

References

  1. D. C. Barnett, R. G. Halburd, R. J. Korhonen, and W. Morgan, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 3, 457-474. https://doi.org/10.1017/S0308210506000102
  2. G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Var. Theory Appl. 37 (1998), no. 1-4, 185-193. https://doi.org/10.1080/17476939808815132
  3. F. Gross, Factorization of meromorphic functions and some open problems, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), pp. 51-67. Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977.
  4. F. Gross and C. F. Osgood, Entire functions with common preimages, in: Factorization theory of meromorphic functions and related topics, pp. 19-24, Lecture Notes in Pure and Appl. Math., 78, Dekker, New York,
  5. X. H. Hua, A unicity theorem for entire functions, Bull. London Math. Soc. 22 (1990), no. 5, 457-462. https://doi.org/10.1112/blms/22.5.457
  6. I. Laine and C. C. Yang, Clunie theorem for difference and q-difference polynomials, J. London. Math. Soc. (2) 76 (2007), no. 3, 556-566. https://doi.org/10.1112/jlms/jdm073
  7. R. Nevanlinna, Le theoreme de Picard-Borel et la theorie des fonctions meromorphes, Gauthiers-Villars, Paris, 1929.
  8. X. G. Qi, K. Liu, and L. Z. Yang, Value sharing results of a meromorphic function f(z) and f(qz), Bull. Korean. Math. Soc. 48 (2011), no. 6, 1235-1243. https://doi.org/10.4134/BKMS.2011.48.6.1235
  9. X. G. Qi and L. Z. Yang, Share sets of q-difference of meromorphic functions, to appear in Math. Slovaca.
  10. K. Shibazaki, Unicity theorem for entire functions of finite order, Mem. Nat. Defense Acad. (Japan) 21 (1981), no. 3, 67-71.
  11. C. C. Yang, On two entire functions which together with their first derivative have the same zeros, J. Math. Anal. Appl. 56 (1976), no. 1, 1-6. https://doi.org/10.1016/0022-247X(76)90002-0
  12. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, 2003.
  13. L. Z. Yang and J. L. Zhang, Non-existence of meromorphic solution 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, A question of Gross and the uniqueness of entire functions, Nagoya Math. J. 138 (1995), 169-177. https://doi.org/10.1017/S0027763000005225
  15. H. X. Yi, Unicity theorems for meromorphic or entire functions II, Bull. Austral. Math. Soc. 52 (1995), no. 2, 215-224. https://doi.org/10.1017/S0004972700014635
  16. H. X. Yi, Unicity theorems for meromorphic or entire functions III, Bull. Austral. Math. Soc. 53 (1996), no. 1, 71-82. https://doi.org/10.1017/S0004972700016737
  17. H. X. Yi, Meromorphic functions that share one or two values II, Kodai Math. J. 22, (1999), no. 2, 264-272. https://doi.org/10.2996/kmj/1138044046
  18. H. X. Yi and L. Z. Yang, Meromorphic functions that share two sets, Kodai Math. J. 20 (1997), no. 2, 127-134. https://doi.org/10.2996/kmj/1138043751
  19. X. B. Zhang, Uniqueness of meromorphic functions sharing a small function and some normality criteria, Doctoral Dissertation, Shandong University, 2011.
  20. J. L. Zhang and R. Korhonen, On the Nevanlinna characteristic of f(qz) and its applications, J. Math. Anal. Appl. 369 (2010), no. 2, 537-544. https://doi.org/10.1016/j.jmaa.2010.03.038