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On the zeros of the second derivative

Published online by Cambridge University Press:  14 November 2011

J. K. Langley
J. K. Langley
Affiliation:
Department of Mathematics, University of Nottingham, Nottingham, NG7 2RD, U.K.
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Abstract

Suppose that f is meromorphic of finite order in the plane, and that f″ has only finitely many zeros. We prove a strong estimate for the frequency of distinct poles of f. In particular, if the poles of f have bounded multiplicities, then f has only finitely many poles.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 2 , 1997 , pp. 359 - 368
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Bergweiler, W. and Eremenko, A.. On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoamericana 11 (1995), 355–73.CrossRefGoogle Scholar
2Edrei, A. and Fuchs, W. H. J.. Valeurs déficientes et valeurs asymptotiques des fonctions méromorphes. Comment. Math. Helv. 33 (1959), 258–95.CrossRefGoogle Scholar
3Eremenko, A.. Meromorphic functions with small ramification. Indiana Univ. Math. J. 42 (1994), 1193–218.CrossRefGoogle Scholar
4Eremenko, A., Langley, J. K. and Rossi, J.. On the zeros of meromorphic functions of the form J. Anal. Math. 62 (1994), 271–86.CrossRefGoogle Scholar
5Frank, G.. Eine Vermutung von Hayman fiber Nullstellen meromorpher Funktionen. Math. Z. 149 (1976), 29–36.CrossRefGoogle Scholar
6Frank, G. and Hellerstein, S.. On the meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients. Proc. London Math. Soc. (3) 53 (1986), 407–28.CrossRefGoogle Scholar
7Frank, G., Hennekemper, W. and Polloczek, G.. Über die Nullstellen meromorpher Funktionen and ihrer Ableitungen. Math. Ann. 225 (1977), 145–54.CrossRefGoogle Scholar
8Hayman, W. K.. Picard values of meromorphic functions and their derivatives. Ann. of Math. 70 (1959), 9–42.CrossRefGoogle Scholar
9Hayman, W. K.. Meromorphic Functions (Oxford: Clarendon Press, 1964).Google Scholar
10Hayman, W. K.. Subharmonic Functions Vol. 2 (London: Academic Press, 1989).Google Scholar
11Keogh, F. R.. A property of bounded schlicht functions. J. London Math. Soc. 29 (1954), 379–82.CrossRefGoogle Scholar
12Langley, J. K.. Proof of a conjecture of Hayman concerning f and f“. J. London Math. Soc. (2) 48 (1993), 500–14.CrossRefGoogle Scholar
13Langley, J. K.. A lower bound for. the numbers of zeros of a meromorphic function and its second derivative. Proc. Edinburgh Math. Soc. 39 (1996), 171–85.CrossRefGoogle Scholar
14Langley, J. K.. The zeros of the first two derivatives of a meromorphic function. Proc. Amer. Math. Soc. 124 (1996), 2439–41.CrossRefGoogle Scholar
15Langley, J. K.. The zeros of the second derivative of a meromorphic function. XVlth Rolf Nevanlinna Colloquium, eds Laine, /Martio, (Berlin: Walter de Gruyter, 1996).Google Scholar
16Langley, J. K. and Shea, D. F.. On multiple points of meromorphic functions. J. London Math. Soc. (to appear).Google Scholar
17Mues, E.. Uber eine Defekt- und Verzweigungsrelation fur die Ableitung meromorpher Funktionen. Manuscripta Math. 5 (1971), 275–97.CrossRefGoogle Scholar
18Nevanlinna, R.. Eindeutige analytische Funktionen, 2. Auflage (Berlin: Springer, 1953).CrossRefGoogle Scholar
19Pólya, G.. Über die Nullstellen sukzessiver Derivierten. Math. Z. 12 (1922), 36–60.CrossRefGoogle Scholar
20Weitsman, A.. A theorem on Nevanlinna deficiencies. Acta Math. 128 (1972), 41–52.CrossRefGoogle Scholar