Abstract and Applied Analysis

On the Deficiencies of Some Differential-Difference Polynomials

Xiu-Min Zheng and Hong Yan Xu

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Abstract

The characteristic functions of differential-difference polynomials are investigated, and the result can be viewed as a differential-difference analogue of the classic Valiron-Mokhon’ko Theorem in some sense and applied to investigate the deficiencies of some homogeneous or nonhomogeneous differential-difference polynomials. Some special differential-difference polynomials are also investigated and these results on the value distribution can be viewed as differential-difference analogues of some classic results of Hayman and Yang. Examples are given to illustrate our results at the end of this paper.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 378151, 12 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412607233

Digital Object Identifier
doi:10.1155/2014/378151

Mathematical Reviews number (MathSciNet)
MR3176739

Zentralblatt MATH identifier
07022257

Citation

Zheng, Xiu-Min; Xu, Hong Yan. On the Deficiencies of Some Differential-Difference Polynomials. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 378151, 12 pages. doi:10.1155/2014/378151. https://projecteuclid.org/euclid.aaa/1412607233


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References

  • W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964.
  • I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, Germany, 1993.
  • C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 2003.
  • Y.-M. Chiang and S.-J. Feng, “On the Nevanlinna characteristic of f(z+$\eta $) and difference equations in the complex plane,” Ramanujan Journal, vol. 16, no. 1, pp. 105–129, 2008.
  • R. G. Halburd and R. J. Korhonen, “Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 477–487, 2006.
  • R. G. Halburd, R. J. Korhonen, and K. Toghe, “Holomorphic curves with shift-invariant hyper-plane čommentComment on ref. [8?]: Please update the information of this reference, if possible.preimages,” Transactions of the American Mathematical Society. In press, http:// arxiv.org/abs/0903.3236.
  • M. J. Ablowitz, R. Halburd, and B. Herbst, “On the extension of the Painlevé property to difference equations,” Nonlinearity, vol. 13, no. 3, pp. 889–905, 2000.
  • W. Bergweiler and J. K. Langley, “Zeros of differences of meromorphic functions,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 142, no. 1, pp. 133–147, 2007.
  • Z. X. Chen, “Complex oscillation of meromorphic solutions for the Pielou logistic equation,” Journal of Difference Equations and Applications, vol. 19, no. 11, pp. 1795–1806, 2013.
  • Z. X. Chen and K. H. Shon, “Fixed points of meromorphic solutions for some difference equations,” Abstract and Applied Analysis, vol. 2013, Article ID 496096, 7 pages, 2013.
  • K. Ishizaki and N. Yanagihara, “Wiman-Valiron method for difference equations,” Nagoya Mathematical Journal, vol. 175, pp. 75–102, 2004.
  • I. Laine and C.-C. Yang, “Clunie theorems for difference and q-difference polynomials,” Journal of the London Mathematical Society, vol. 76, no. 3, pp. 556–566, 2007.
  • I. Laine and C. C. Yang, “Value distribution of difference polynomials,” Proceedings of the Japan Academy A, vol. 83, no. 8, pp.148–151, 2007.
  • C.-C. Yang and I. Laine, “On analogies between nonlinear difference and differential equations,” Proceedings of the Japan Academy A, vol. 86, no. 1, pp. 10–14, 2010.
  • R. R. Zhang and Z. B. Huang, “Results on difference analogues of Valiron-Mokhon'ko theorem,” Abstract and Applied Analysis, vol. 2013, Article ID 273040, 6 pages, 2013.
  • X. M. Zheng and Z. X. Chen, “On deficiencies of some difference polynomials,” Acta Mathematica Sinica, vol. 54, no. 6, pp. 983–992, 2011 (Chinese).
  • X. M. Zheng and Z. X. Chen, “On the value distribution of some difference polynomials,” Journal of Mathematical Analysis and Applications, vol. 397, no. 2, pp. 814–821, 2013.
  • G. Valiron, “Sur la dérivée des fonctions algébroides,” Bulletin de la Société Mathématique de France, vol. 59, pp. 17–39, 1931.
  • A. Z. Mokhon'ko and V. D. Mokhon'ko, “Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions and their applications to differential equations,” Sibirskii Matematicheskii Zhurnal, vol. 15, pp. 1305–1322, 1974 (Russian).
  • W. K. Hayman, “Picard values of meromorphic functions and their derivatives,” Annals of Mathematics, vol. 70, no. 2, pp. 9–42, 1959.
  • C.-C. Yang, “On deficiencies of differential polynomials,” Mathematische Zeitschrift, vol. 116, no. 3, pp. 197–204, 1970.
  • C.-C. Yang, “On deficiencies of differential polynomials, II,” Mathematische Zeitschrift, vol. 125, no. 2, pp. 107–112, 1972.
  • A. A. Gol'dberg and I. V. Ostrovskii, The Distribution of Values of Meromorphic Functions, Nauka, Moscow, Russia, 1970, (Russian).
  • E. Mues and N. Steinmetz, “The theorem of Tumura-Clunie for meromorphic functions,” Journal of the London Mathematical Society, vol. 23, no. 2, pp. 113–122, 1981. \endinput