The Zeros of a Certain Homogeneous Difference Polynomials of Meromorphic Functions

Affiliation(s)

Department of Mathematics, Southwest University of Science and Technology, Mianyang, China.

Department of Material Science and Engineer, Southwest University of Science and Technology, Mianyang, China.

Department of Mathematics, Southwest University of Science and Technology, Mianyang, China.

Department of Material Science and Engineer, Southwest University of Science and Technology, Mianyang, China.

ABSTRACT

Let *f*(*z*) be a function transcendental and meromorphic in the plane of growth order less than 1. This paper focuses on discuss and estimate the number of the zeros of a certain homogeneous difference polynomials of degree *k* in *f*(*z*), and obtains that this certain homogeneous difference polynomials has infinitely many zeros.

Cite this paper

Q. Lu and Q. Liao, "The Zeros of a Certain Homogeneous Difference Polynomials of Meromorphic Functions,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 99-104. doi: 10.4236/apm.2013.31012.

Q. Lu and Q. Liao, "The Zeros of a Certain Homogeneous Difference Polynomials of Meromorphic Functions,"

References

[1] J. M. Whittaker, “Interpolatory Function Theory,” Cambridge Tracts in Mathematics and Mathematical Physics, No. 33, Cambridge University Press, New York, 1935, p. 52.

[2] M. Ablowitz, R. G. Halburd and B. Herbst, “On the Extension of Painleve Property to Difference Equations,” Nonlinearty, Vol. 13, No. 3, 2000, pp. 889-905. doi:10.1088/0951-7715/13/3/321

[3] R. G. Halburd and R. 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, 2006, pp. 477- 487. doi:10.1016/j.jmaa.2005.04.010

[4] R. G. Halburd and R. Korhonen, “Nevanlinna Theory for the Difference Operator,” Annales Academiae Scientiarum Fennicae Mathematica, Vol. 31, No. 2, 2006, pp. 463-478.

[5] I. Laine and C. C. Yang, “Value Distribution of Difference Polynomials,” Proceedings of the Japan Academy, Vol. 83, No. 8, 2007, pp. 148-151. doi:10.3792/pjaa.83.148

[6] Y. M. Chiang and S. J. Feng, “On the Nevanlinna Characteristic of f(z + c) and Difference Equations in the Complex Plane,” The Ramanujan Journal, Vol. 16, No. 1, 2008, pp. 105-129. doi:10.1007/s11139-007-9101-1

[7] W. Bergweiler and J. K. Langley, “Zeros of Differences of Meromorphic Functions,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 142, No. 1, 2007, pp. 133-147. doi:10.1017/S0305004106009777

[8] Z. X. Chen and K. H. Shon, “On Zeros and Fixed Points of Difference of Meromorphic Functions,” Journal of Mathematical Analysis and Applications, Vol. 344, No. 1, 2008, pp. 373-383. doi:10.1016/j.jmaa.2008.02.048

[9] Z. X. Chen and K. H. Shon, “Estimates for the Zeros of Difference of Meromorphic Functions,” Science China, Series A, Vol. 52, No. 11, 2009, pp. 2447-2458. doi:10.1007/s11425-009-0159-7

[1] J. M. Whittaker, “Interpolatory Function Theory,” Cambridge Tracts in Mathematics and Mathematical Physics, No. 33, Cambridge University Press, New York, 1935, p. 52.

[2] M. Ablowitz, R. G. Halburd and B. Herbst, “On the Extension of Painleve Property to Difference Equations,” Nonlinearty, Vol. 13, No. 3, 2000, pp. 889-905. doi:10.1088/0951-7715/13/3/321

[3] R. G. Halburd and R. 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, 2006, pp. 477- 487. doi:10.1016/j.jmaa.2005.04.010

[4] R. G. Halburd and R. Korhonen, “Nevanlinna Theory for the Difference Operator,” Annales Academiae Scientiarum Fennicae Mathematica, Vol. 31, No. 2, 2006, pp. 463-478.

[5] I. Laine and C. C. Yang, “Value Distribution of Difference Polynomials,” Proceedings of the Japan Academy, Vol. 83, No. 8, 2007, pp. 148-151. doi:10.3792/pjaa.83.148

[6] Y. M. Chiang and S. J. Feng, “On the Nevanlinna Characteristic of f(z + c) and Difference Equations in the Complex Plane,” The Ramanujan Journal, Vol. 16, No. 1, 2008, pp. 105-129. doi:10.1007/s11139-007-9101-1

[7] W. Bergweiler and J. K. Langley, “Zeros of Differences of Meromorphic Functions,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 142, No. 1, 2007, pp. 133-147. doi:10.1017/S0305004106009777

[8] Z. X. Chen and K. H. Shon, “On Zeros and Fixed Points of Difference of Meromorphic Functions,” Journal of Mathematical Analysis and Applications, Vol. 344, No. 1, 2008, pp. 373-383. doi:10.1016/j.jmaa.2008.02.048

[9] Z. X. Chen and K. H. Shon, “Estimates for the Zeros of Difference of Meromorphic Functions,” Science China, Series A, Vol. 52, No. 11, 2009, pp. 2447-2458. doi:10.1007/s11425-009-0159-7