VALUE DISTRIBUTION OF DIFFERENCE OPERATOR ON MEROMORPHIC FUNCTIONS

• Ding, Jie (School of Mathematics Taiyuan University of Technology) ;
• Yang, Lian-Zhong (School of Mathematics Shandong University)
• Received : 2012.02.03
• Published : 2014.09.30

Abstract

In this paper, we investigate the value distribution of the difference operator on meromorphic functions, and obtain a difference analogue of a theorem of Hayman on meromorphic functions.

References

1. D. Barnett, R. G. Halburd, W. Morga, and R. J. Korhonen, Nevanlinna theory for the qq-difference operator and meromorphic solutions of qq-difference equations, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 457-474. https://doi.org/10.1017/S0308210506000102
2. W. Bergweiler and J. K. Langley, Zeros of difference of meromorphic functions, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 133-147. https://doi.org/10.1017/S0305004106009777
3. Z. X. Chen, On value distribution of difference polynomial of meromorphic functions, Abstr. Appl. Anal. 2011 (2011), 1-9.
4. Y. M. Chiang and S. J. Feng, On the growth of logarithmic differences, difference quo-tients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3767-3791. https://doi.org/10.1090/S0002-9947-09-04663-7
5. Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic f($z+{\eta}$) and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105-129. https://doi.org/10.1007/s11139-007-9101-1
6. J. Grahl, Differential polynomial with dilations in the argument and normal families, Monatsh. Math. 162 (2011), no. 4, 429-452. https://doi.org/10.1007/s00605-009-0186-z
7. R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the Logarithmic Derivative with application to difference equations, J. Math. Anal. Appl. 314 (2006), no. 2, 477-487. https://doi.org/10.1016/j.jmaa.2005.04.010
8. R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 463-478.
9. R. G. Halburd and R. J. Korhonen, Finite-order meromorphic solutions and the discrete Painleve equations, Proc. Lond. Math. Soc. 94 (2007), no. 2, 443-474. https://doi.org/10.1112/plms/pdl012
10. W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. 70 (1959), 9-42. https://doi.org/10.2307/1969890
11. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964.
12. J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge, Complex difference equations of Malmquist type, Comput. Methods Funct. Theory 1 (2001), no. 1, 27-39. https://doi.org/10.1007/BF03320974
13. J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and L. J. Zhang, Value sharing results for shifts of meromorphic function and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009), no. 1, 352-363. https://doi.org/10.1016/j.jmaa.2009.01.053
14. K. Ishizaki and N. Yanagihara, Wiman-Valiron method for difference equations, Nagoya Math. J. 175 (2004), 75-102. https://doi.org/10.1017/S0027763000008916
15. I. Laine, Nevanlinna Theory and Complex Differential Equation, Walter de Gruyter, Berlin-New York, 1993.
16. I. Laine and C. C. Yang, Clunie theorems for difference q-difference polynomials, J. Lond. Math. Soc. 76 (2007), no. 3, 556-566. https://doi.org/10.1112/jlms/jdm073
17. I. Laine and C. C. Yang, Value distribution of difference polynomials, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 8, 148-151. https://doi.org/10.3792/pjaa.83.148
18. K. Liu and I. Laine, A note on value distribution of difference polynomial, Bull. Aust. Math. Soci. 81 (2010), no. 3, 353-360. https://doi.org/10.1017/S000497270900118X
19. K. Liu and L. Z. Yang, Value distribution of the difference operator, Arch. Math. 92 (2009), no. 3, 270-278. https://doi.org/10.1007/s00013-009-2895-x
20. A. I. Markushevich, Theory of Functions of a Complex Variable. Vol. 2, Englewood Cliffs, N. J. Prentice-Hall, 1965.
21. J. M. Qi, J. Ding, and T. Y. Zhu, Some results about a special nonlinear difference equation and uniqueness of diffrence polynomial, J. Inequal. Appl. 2011 (2011), 50, 1-10. https://doi.org/10.1186/1029-242X-2011-1
22. X. G. Qi and K. Liu, Uniqueness and value distribution of differences of entire functions, J. Math. Anal. Appl. 379 (2011), no. 1, 180-187. https://doi.org/10.1016/j.jmaa.2010.12.031
23. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, 2004.
24. 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

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