• Luong, Thi Tuyet (Department of Mathematics National University of Civil Engineering) ;
  • Nguyen, Dang Tuyen (Department of Mathematics National University of Civil Engineering) ;
  • Pham, Duc Thoan (Department of Mathematics National University of Civil Engineering)
  • Received : 2016.11.20
  • Accepted : 2017.08.11
  • Published : 2018.01.31


In this paper, we show the Second Main Theorems for zero-order meromorphic mapping of ${\mathbb{C}}^m$ into ${\mathbb{P}}^n({\mathbb{C}})$ intersecting hyperplanes in subgeneral position without truncated multiplicity by considering the p-Casorati determinant with $p{\in}{\mathbb{C}}^m$ instead of its Wronskian determinant. As an application, we give some unicity theorems for meromorphic mapping under the growth condition "order=0". The results obtained include p-shift analogues of the Second Main Theorem of Nevanlinna theory and Picard's theorem.


  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.
  2. T. B. Cao, Difference analogues of the second main theorem for meromorphic functions in several complex variables, Math. Nachr. 287 (2014), no. 5-6, 530-545.
  3. T. B. Cao and R. Korhonen, A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables, J. Math. Anal. Appl. 444 (2016), no. 2, 1114-1132.
  4. H. Cartan, Sur les zeros des combinaisons lineaires de pfonctions holomorphes donnees, Mathematica Cluj 7 (1933), 531.
  5. H. Fujimoto, The uniqueness problem of meromorphic maps into the complex projective space, Nagoya Math. J. 58 (1975), 1-23.
  6. H. Fujimoto, Nonintegrated defect relation for meromorphic maps of complete Kahler manifolds into ${\mathbb{P}}^{N_1}({\mathbb{C}}){\times}{\cdot}{\cdot}{\cdot}{\times}{\mathbb{P}}^{N_k}({\mathbb{C}})$, Japan. J. Math. (N.S.) 11 (1985), no. 2, 233-264.
  7. R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 463-478.
  8. R. Halburd, R. Korhonen, and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4267-4298.
  9. W. K. Hayman, On the characteristic of functions meromorphic in the plane and of their integrals, Proc. Lond. Math. Soc. (3) 14 (1965), 93-128.
  10. P-C. Hu, P. Li, and C-C. Yang, Unicity of Meromorphic Mappings, Vol. 1 of Advances in Complex Analysis and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
  11. Z.-B. Huang, Value distribution and uniqueness on q-differences of meromorphic functions, Bull. Korean Math. Soc. 50 (2013), no. 4, 1157-1171.
  12. R. Korhonen, A difference Picard theorem for meromorphic functions of several variables, Comput. Methods Funct. Theory 12 (2012), no. 1, 343-361.
  13. E. I. Nochka, On the theory of meromorphic functions, Sov. Math. Dokl. 27 (1983), 377-381.
  14. J. Noguchi, A note on entire pseudo-holomorphic curves and the proof of Cartan-Nochka's theorem, Kodai Math. J. 28 (2005), no. 2, 336-346.
  15. X. Qi, K. Liu, and L. Yang, Value results of a meromorphic function f(z) and f(qz), Bull. Korean Math. Soc. 48 (2011), no. 6, 1235-1243.
  16. Z. T. Wen, The q-difference theorems for meromorphic functions of several variables, Abstr. Appl. Anal. 2014 (2014), ID 736021, 6 pp.
  17. Z.-T. Wen and Z. Ye, Wimam-Valiron theorem for q-difference, Annales AcademiaeSci-entiarum FennicaeMathematica 41 (2016), 305-312.
  18. P. M.Wong, H. F. Law, and P. P. W.Wong, A second main theorem on $P^n$ for difference operator, Sci. China Ser. A 52 (2009), no. 12, 2751-758.
  19. J. Zhang and R. Korhonen, On the Nevanlinna characteristic of f(qz) and its applications , J. Math. Anal. Appl. 369 (2010), no. 2, 537-544.