Acta mathematica scientia, Series B >
ZEROS AND FIXED POINTS OF DIFFERENCE OPERATORS OF MEROMORPHIC FUNCTIONS
Received date: 2011-01-04
Online published: 2013-05-20
Supported by
The second author was supported by the NSF of Shandong Province, China (ZR2010AM030) and the NNSF of China (11171013 & 11041005).
Let f be a transcendental meromorphic function and Δf(z) = f(z + 1) − f(z). A number of results are proved concerning the existences of zeros and fixed points of Δf(z) and Δf(z)/f(z) when f(z) is of order σ(f)=1. Examples show that some of the results are sharp.
Key words： Entire functions; meromorphic functions; complex difference; fixed points; Zeros
CUI Wei-Wei , YANG Lian-Zhong . ZEROS AND FIXED POINTS OF DIFFERENCE OPERATORS OF MEROMORPHIC FUNCTIONS[J]. Acta mathematica scientia, Series B, 2013 , 33(3) : 773 -780 . DOI: 10.1016/S0252-9602(13)60037-5
[1] Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964
[2] Yi H X, Yang C C. The Uniqueness Theory of Meromorphic Functions. Beijing: Science Press, 1995 (in Chinese)
[3] Yang L. Value Distribution Theory and New Research. Beijing: Science Press, 1982 (in Chinese)
[4] Bergweiler W, Langley J K. Zeros of differences of meromorphic functions. Math Proc Cambridge Philos Soc, 2007, 142: 133–147
[5] Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin: W de Gruyter, 1993
[6] Whittaker J M. Interpolatory Function Theory. Cambridge Tracts in Math and Math Phys. Vol 33. Cam-bridge University Press, 1935
[7] Bergweiler W, Eremenko A. On the singularities of the inverse to a meromorphic function of finite order. Rev Mat Iberoamericana, 1995, 11: 355–373
[8] Eremenko A, Langley J K, Rossi J. On the zeros of meromorphic functions of the form 1P k=1akz−zk. J Anal
Math, 1994, 62: 271–286
[9] Hinchliffe J D. The Bergweiler-Eremenko theorem for the finite lower order. Results Math, 2003, 43: 121–128
[10] Chen Z X, Shon K H. On zeros of fixed points of differences of meromorphic functions. J Anal Math Appl, 2008, 344: 373–383
[11] Hayman W K. Slowly growing integral and subharmonic functions. Comment Math Helv, 1960, 34: 75–84
[12] Halburd R G, Korhonen R. Nevanlinna theory for the difference operator. Ann Acad Sci Fenn Math, 2006, 31: 463–478
[13] Halburd R G, Korhonen R. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J Anal Math Appl, 2006, 314: 477–487
[14] Chen Z X. On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations. Sci China Ser A, 2011, 54(10): 2121–2133
[15] Yang L Z, Zhang J L. On some results of Yi and their applications in the functional equations. Acta Mathematica Scientia, 2010, 30B(5): 1649–1660
[16] Lin W, Yi H. Meromorphic functions sharing two finite sets. Acta Mathematica Scientia, 2007, 27B(4): 845–851
/
| 〈 |
|
〉 |