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Zeros of differences of meromorphic functions

Published online by Cambridge University Press:  12 February 2007

WALTER BERGWEILER
Affiliation:
Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D-24098 Kiel, Germany. e-mail: [email protected]
J. K. LANGLEY
Affiliation:
School of Mathematical Sciences, University of Nottingham, NG7 2RD. e-mail: [email protected]

Abstract

Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = Δf(z) = f(z + 1) − f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f. The results may be viewed as discrete analogues of existing theorems on the zeros of f' and f'/f.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1] Ablowitz, M., Halburd, R. G. and Herbst, B.. On the extension of the Painlevé property to difference equations. Nonlinearity 13 (2000), 889905.Google Scholar
[2] Anderson, J. M. and Clunie, J.. Slowly growing meromorphic functions. Comment. Math. Helv. 40 (1966), 267280.CrossRefGoogle Scholar
[3] Barry, P. D.. On a theorem of Kjellberg. Quar t. J. Math. Oxford (2) 15 (1964), 179191.CrossRefGoogle Scholar
[4] Bergweiler, W. and Eremenko, A.. On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoamericana 11 (1995), 355373.Google Scholar
[5] Chiang, Y. M. and Feng, S. J.. On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane. Ramanujan J., to appear.Google Scholar
[6] Clunie, J., Eremenko, A. and Rossi, J.. On equilibrium points of logarithmic and Newtonian potentials. J. London Math. Soc. (2) 47 (1993), 309320.CrossRefGoogle Scholar
[7] Eremenko, A., Langley, J. K. and Rossi, J.. On the zeros of meromorphic functions of the form . J. Anal. Math. 62 (1994), 271286.Google Scholar
[8] Gol'dberg, A. A. and Ostrowski, I. V.. Distribution of Values of Meromorphic Functions (Nauka, 1970).Google Scholar
[9] Gundersen, G.. Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. London Math. Soc. (2) 37 (1988), 88104.CrossRefGoogle Scholar
[10] Halburd, R. G. and Korhonen, R.. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314 (2006), 477487.Google Scholar
[11] Halburd, R. G. and Korhonen, R.. Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math., to appear.Google Scholar
[12] Hayman, W. K.. Slowly growing integral and subharmonic functions. Comment. Math. Helv. 34 (1960), 7584.Google Scholar
[13] Hayman, W. K.. Meromorphic Functions (Oxford at the Clarendon Press, 1964).Google Scholar
[14] Hayman, W. K.. The local growth of power series: a survey of the Wiman–Valiron method. Canad. Math. Bull. 17 (1974), 317358.Google Scholar
[15] Hayman, W. K.. Subharmonic Functions Vol. 2 (Academic Press, 1989).Google Scholar
[16] Heittokangas, J., Korhonen, R., Laine, I., Rieppo, J. and Tohge, K.. Complex difference equations of Malmquist type. Comput. Methods Funct. Theory 1 (2001), 2739.Google Scholar
[17] Hinchliffe, J. D.. The Bergweiler–Eremenko theorem for finite lower order. Results. Math. 43 (2003), 121128.Google Scholar
[18] Ishizaki, K. and Yanagihara, N.. Wiman–Valiron method for difference equations. Nagoya Math. J. 175 (2004), 75102.Google Scholar
[19] Jank, G. and Volkmann, L.. Einführung in die Theorie der Ganzen und Meromorphen Funktionen mit Anwendungen auf Differentialgleichungen (Birkhäuser, 1985).Google Scholar
[20] Miles, J. and Rossi, J.. Linear combinations of logarithmic derivatives of entire functions with applications to differential equations. Pacific J. Math. 174 (1996), 195214.Google Scholar
[21] Tsuji, M.. Potential Theory in Modern Function Theory (Maruzen, 1959).Google Scholar
[22] Valiron, G.. Lectures on the General Theory of Integral Functions (Edouard Privat, Toulouse, 1923).Google Scholar
[23] Whittaker, J. M.. Interpolatory Function Theory. Cambridge Tracts in Mathematics and Mathematical Physics no. 33 (Cambridge University Press, 1935).Google Scholar