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On zeros and fixed points of differences of meromorphic functions. (English) Zbl 1144.30012
This article is devoted to investigating zeros and fixed points of differences of entire and meromorphic functions in the complex plane, extending previous results due to Bergweiler and Langley [W. Bergweiler and J. K. Langley, Math. Proc. Camb. Philos. Soc. 142, No. 1, 133–147 (2007; Zbl 1114.30028)]. Denoting, for a transcendental meromorphic function \(f\), \(\Delta f(z):=f(z+c)-f(z)\), \(\Delta^{n+1}f(z):=\Delta^{n}f(z+c)-\Delta^{n}f(z)\), the results obtained are treating differences and divided differences \(G(z):=\Delta f(z)/f(z)\), \(G_{n}(z):=\Delta^{n}(z)/f(z)\) of \(f\). The key results in this paper are as follows: (1) \(G_{n}(z)\) has infinitely many zeros and infinitely many fixed points, provided \(f\) is entire, \(c=1\) and \(\rho (f)<1/2\) and \(\rho (f)\neq j/n\), \(j=1,\ldots ,[n/2]\). (2) \(G(z)\) has infinitely many zeros and infinitely many fixed points, whenever \(f\) is entire with \(\rho (f)<1\), and either \(f\) has at most finitely many zeros whose difference is \(=c\) or \(\liminf_{j\rightarrow\infty}| z_{j+1}/z_{j}| =L>1\), where \(\{z_{j}\}\) is the zero-sequence of \(f\), arranged according to increasing moduli. (3) A similar result holds, if \(f\) is entire of order \(\rho (f)=1\) and with infinitely many zeros having the exponent of convergence \(\lambda (f)<1\). (4) As for the case of \(f\) meromorphic, a result similar to (2) follows by invoking corresponding conditions for the poles of \(f\) as well. (5) Given a positive, non-decreasing function \(\varphi :[1,\infty )\rightarrow [0,\infty)\) with \(\lim_{r\rightarrow\infty}\varphi (r)=\infty\), there exists \(f\) transcendental meromorphic such that \(\limsup_{r\rightarrow\infty}(T(r,f)/r)<\infty\), \(\limsup_{r\rightarrow\infty}(T(r,f)/\varphi (r)\log r)<\infty\) and that \(\Delta f(z)\) has one fixed point only. The proofs rely on standard properties of meromorphic functions, Wiman-Valiron theory and some Nevanlinna theory.

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Full Text: DOI
[1] Ablowitz, M.; Halburd, R.G.; Herbst, B., On the extension of PainlevĂ© property to difference equations, Nonlinearity, 13, 889-905, (2000) · Zbl 0956.39003
[2] Bergweiler, W.; Langley, J.K., Zeros of differences of meromorphic functions, Math. proc. Cambridge philos. soc., 142, 133-147, (2007) · 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, 355-373, (1995) · Zbl 0830.30016
[4] Y.M. Chiang, S.J. Feng, On the Nevanlinna characteristic of \(f(z + \eta)\) and difference equations in the complex plane, Ramanujan J., in press · Zbl 1152.30024
[5] Clunie, J.; Eremenko, A.; Rossi, J., On equilibrium points of logarithmic and Newtonian potentials, J. London math. soc., 47, 2, 309-320, (1993) · Zbl 0797.31002
[6] Eremenko, A.; Langley, J.K.; Rossi, J., On the zeros of meromorphic functions of the form \(\sum_{k = 1}^\infty \frac{a_k}{z - z_k}\), J. anal. math., 62, 271-286, (1994) · Zbl 0818.30020
[7] Gundersen, G., Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London math. soc., 37, 2, 88-104, (1988) · Zbl 0638.30030
[8] Halburd, R.G.; Korhonen, R., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. math. anal. appl., 314, 477-487, (2006) · Zbl 1085.30026
[9] Halburd, R.G.; Korhonen, R., Nevanlinna theory for the difference operator, Ann. acad. sci. fenn. math., 31, 463-478, (2006) · Zbl 1108.30022
[10] Hayman, W.K., Meromorphic functions, (1964), Clarendon Press Oxford · Zbl 0115.06203
[11] Hayman, W.K., The local growth of power series: A survey of the wiman – valiron method, Canad. math. bull., 17, 317-358, (1974) · Zbl 0314.30021
[12] Hayman, W.K., Slowly growing integral and subharmonic functions, Comment. math. helv., 34, 75-84, (1960) · Zbl 0123.26702
[13] Heittokangas, J.; Korhonen, R.; Laine, I.; Rieppo, J.; Hohge, K., Complex difference equations of Malmquist type, Comput. methods funct. theory, 1, 27-39, (2001) · Zbl 1013.39001
[14] Hinchliffe, J.D., The bergweiler – eremenko theorem for finite lower order, Results math., 43, 121-128, (2003) · Zbl 1036.30022
[15] Ishizaki, K.; Yanagihara, N., Wiman – valiron method for difference equations, Nagoya math. J., 175, 75-102, (2004) · Zbl 1070.39002
[16] Laine, I., Nevanlinna theory and complex differential equations, (1993), W. de Gruyter Berlin
[17] Whittaker, J.M., Interpolatory function theory, Cambridge tracts in math. and math. phys., vol. 33, (1935), Cambridge University Press · Zbl 0012.15503
[18] Yang, Lo, Value distribution theory and new research, (1982), Science Press Beijing, (in Chinese) · Zbl 0633.30029
[19] Yi, Hong-Xun; Yang, Chung-Chun, The uniqueness theory of meromorphic functions, (1995), Science Press Beijing, (in Chinese) · Zbl 1070.30011
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