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Article

Keywords:
meromorphic function; difference; divided difference; zero; fixed point
Summary:
Let $f$ be a transcendental meromorphic function. We propose a number of results concerning zeros and fixed points of the difference $g(z)=f(z+c)-f(z)$ and the divided difference $g(z)/f(z)$.
References:
[1] Ablowitz, M., Halburd, R. G., Herbst, B.: On the extension of Painlevé property to difference equations. Nonlinearity 13 (2000), 889-905. DOI 10.1088/0951-7715/13/3/321 | MR 1759006
[2] Bergweiler, W., Langley, J. K.: Zeros of differences of meromorphic functions. Math. Proc. Camb. Phil. Soc. 142 (2007), 133-147. DOI 10.1017/S0305004106009777 | MR 2296397 | Zbl 1114.30028
[3] Bergweiler, W., Eremenko, A.: On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoamericana 11 (1995), 355-373. DOI 10.4171/RMI/176 | MR 1344897 | Zbl 0830.30016
[4] Chen, Z. X., Shon, K. H.: On zeros and fixed points of differences of meromorphic functions. J. Math. Anal. Appl. 344-1 (2008), 373-383. MR 2416313 | Zbl 1144.30012
[5] Chiang, Y. M., Feng, S. J.: On the Nevanlinna characteristic of $f(z+\eta)$ and difference equations in the complex plane. Ramanujan J. 16 (2008), 105-129. DOI 10.1007/s11139-007-9101-1 | MR 2407244 | Zbl 1152.30024
[6] Clunie, J., Eremenko, A., Rossi, J.: On equilibrium points of logarithmic and Newtonian potentials. J. London Math. Soc. 47-2 (1993), 309-320. MR 1207951 | Zbl 0797.31002
[7] Conway, J. B.: Functions of One Complex Variable. New York, Spring-Verlag. MR 0503901 | Zbl 0887.30003
[8] Eremenko, A., Langley, J. K., Rossi, J.: On the zeros of meromorphic functions of the form $\sum\nolimits_{k=1}^{\infty}{a_k}/(z-z_k)$. J. Anal. Math. 62 (1994), 271-286. DOI 10.1007/BF02835958 | MR 1269209
[9] Gundersen, G.: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. London Math. Soc. 37-2 (1988), 88-104. MR 0921748 | Zbl 0638.30030
[10] Halburd, R. G., Korhonen, R.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314 (2006), 477-487. DOI 10.1016/j.jmaa.2005.04.010 | MR 2185244 | Zbl 1085.30026
[11] Halburd, R. G., Korhonen, R.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31 (2006), 463-478. MR 2248826 | Zbl 1108.30022
[12] Hayman, W. K.: Meromorphic Functions. Oxford, Clarendon Press (1964). MR 0164038 | Zbl 0115.06203
[13] Hayman, W. K.: Slowly growing integral and subharmonic functions. Comment. Math. Helv. 34 (1960), 75-84. DOI 10.1007/BF02565929 | MR 0111839 | Zbl 0123.26702
[14] Heittokangas, J., Korhonen, R., Laine, I., Rieppo, J., Tohge, K.: Complex difference equations of Malmquist type. Comput. Methods Funct. Theory 1 (2001), 27-39. DOI 10.1007/BF03320974 | MR 1931600 | Zbl 1013.39001
[15] Hinchliffe, J. D.: The Bergweiler-Eremenko theorem for finite lower order. Results Math. 43 (2003), 121-128. DOI 10.1007/BF03322728 | MR 1962854 | Zbl 1036.30022
[16] Ishizaki, K., Yanagihara, N.: Wiman-Valiron method for difference equations. Nagoya Math. J. 175 (2004), 75-102. MR 2085312 | Zbl 1070.39002
[17] Laine, I.: Nevanlinna Theory and Complex Differential Equations. Berlin, W. de Gruyter (1993). MR 1207139
[18] Yang, L.: Value Distribution Theory. Beijing, Science Press (1993). MR 1301781 | Zbl 0790.30018
[19] Yang, C. C., Yi, H. X.: Uniqueness Theory of Meromorphic Functions. Dordrecht, Kluwer Academic Publishers Group (2003). MR 2105668 | Zbl 1070.30011
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