Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
 All Title Author Keywords Abstract
 Publish in OALib Journal ISSN: 2333-9721 APC: Only \$99

 Relative Articles Value distribution of certain differential polynomials On Value Distribution of Difference Polynomials of Meromorphic Functions Value Distributions and Uniqueness of Difference Polynomials Value Distribution and Uniqueness Theorems for Difference of Entire and Meromorphic Functions Asymptotic distribution of zeros of polynomials satisfying difference equations The Zeros of a Certain Homogeneous Difference Polynomials of Meromorphic Functions Zero distribution of polynomials satisfying a differential-difference equation An ergodic value distribution of some certain meromorphic functions Asymptotic distribution of zeros of a certain class of hypergeometric polynomials The Zeros of Difference Polynomials of Meromorphic Functions More...

# Value Distribution of Certain Type of Difference Polynomials

 Full-Text   Cite this paper

Abstract:

We investigate the value distribution of difference product , for and , respectively, where is a transcendental entire function of finite order and are constants satisfying . 1. Introduction In this paper, we assume that the reader is familiar with the basic notions of Nevanlinna’s value distribution theory (see [1–3]). The notation is defined to be any quantity satisfying as , possibly outside a set of finite linear measures. In addition, we use the notation to denote the order of growth of the meromorphic function and to denote the exponent of convergence of zeros of . Hayman proved the following theorem in [4]. Theorem 1. Let be a transcendental integral function and let be an integer; then assumes all values except possibly zero infinitely often. Clunie proved that if , then Theorem 1 remains valid. Recently, many papers (see [5–17]) focus on complex difference. They obtain many new results on difference using the value distribution theory of meromorphic functions. In [12], Laine and Yang found a difference analogue of Hayman’s result as follows. Theorem 2. Let be a transcendental entire function of finite order and a nonzero complex constant. Then for , assumes every nonzero value infinitely often. Liu and Yang [14] proved the following theorem. Theorem 3. Let be a transcendental entire function of finite order and let be a nonzero complex constant, . Then for , has infinitely many zeros, where is a polynomial in . Chen [6] proved the following theorem. Theorem 4. Let be a transcendental entire function of finite order and let be a constant satisfying . Set where , and is an integer. Then the following statements hold.(i)If satisfies or has infinitely many zeros, then has infinitely many zeros.(ii)If has only finitely many zeros and , then has only finitely many zeros. It is natural to ask what condition will guarantee that assumes every nonzero and zero value infinitely often, where is a linear th order difference operator with varying shifts, operating on a transcendental entire function of finite order. In this paper, we consider the above question for and , respectively, and obtain the following results. Theorem 5. Let be a transcendental entire function of finite order and let , ？？ be constant satisfying and when . Set , where are integers. Then the following statements hold.(i)If satisfies or has infinitely many zeros, then has infinitely many zeros.(ii)If has only finitely many zeros and , then has only finitely many zeros.(iii) has infinitely many zeros, and , where is a small function of . Remark 6. The result of Theorem 5 may be false if

References

 [1] W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964. [2] I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of De Gruyter Studies in Mathematics, De Gruyter, Berlin, Germany, 1993. [3] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic, 2003. [4] W. K. Hayman, “Picard value of meromorphic funcitons and their derivaties,” Annals of Mathematics, vol. 70, no. 1, pp. 9–42, 1959. [5] 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. [6] Z.-X. Chen, “Value distribution of products of meromorphic functions and their differences,” Taiwanese Journal of Mathematics, vol. 15, no. 4, pp. 1411–1421, 2011. [7] Z. X. Chen, “On value distribution of difference polynomials of meromorphic functions,” Abstract and Applied Analysis, vol. 2011, Article ID 239853, 9 pages, 2011. [8] Y.-M. Chiang and S.-J. Feng, “On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane,” Ramanujan Journal, vol. 16, no. 1, pp. 105–129, 2008. [9] 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. [10] R. G. Halburd and R. J. Korhonen, “Nevanlinna theory for the difference operator,” Annales Academiae Scientiarum Fennicae Mathematica, vol. 31, pp. 463–478, 2006. [11] J. Heittokangas, R. Korgonen, I. Laine, J. Rieppo, and K. Tohge, “Complex difference equations of Malmquist type,” Computational Methods and Function Theory, vol. 1, no. 1, pp. 27–39, 2001. [12] 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. [13] N. Li and L. Z. Yang, “Value distribution of difference and q-difference polynomials,” Advance in Difference Equations, vol. 2013, no. 1, p. 98, 2013. [14] K. Liu and L. Z. Yang, “Value distribution of the difference operator,” Archiv der Mathematik, vol. 92, pp. 270–278, 2009. [15] X.-G. Qi, L.-Z. Yang, and K. Liu, “Uniqueness and periodicity of meromorphic functions concerning the difference operator,” Computers and Mathematics with Applications, vol. 60, no. 6, pp. 1739–1746, 2010. [16] J. Zhang, “Value distribution and shared sets of differences of meromorphic functions,” Journal of Mathematical Analysis and Applications, vol. 367, no. 2, pp. 401–408, 2010. [17] R. R. Zhang and Z. X. Chen, “Value distribution of difference polynomials of meromorphic functions,” Scientia Sinica A, vol. 42, no. 11, pp. 1115–1130, 2012.

Full-Text