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NONDECREASING FUNCTIONS, EXCEPTIONAL SETS AND GENERALIZED BOREL LEMMAS

Part of: Entire and meromorphic functions, and related topics Real functions

Published online by Cambridge University Press:  11 June 2010

R. G. HALBURD  and
R. J. KORHONEN
R. G. HALBURD
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK (email: r.halburd@ucl.ac.uk)
R. J. KORHONEN*
Affiliation:
Department of Physics and Mathematics, University of Eastern Finland, Joensuu Campus, PO Box 111, FI-80101 Joensuu, Finland (email: risto.korhonen@helsinki.fi)
*
✉For correspondence; e-mail: risto.korhonen@helsinki.fi
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Abstract

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According to the classical Borel lemma, any positive nondecreasing continuous function T satisfiesT(r+1/T(r))≤2T(r) outside a possible exceptional set of finite linear measure. This lemma plays an important role in the theory of entire and meromorphic functions, where the increasing function T is either the logarithm of the maximum modulus function, or the Nevanlinna characteristic. As a result, exceptional sets appear throughout Nevanlinna theory, in particular in Nevanlinna’s second main theorem. In this paper, we consider generalizations of Borel’s lemma. Conversely, we consider ways in which certain inequalities can be modified so as to remove exceptional sets. All results discussed are presented from the point of view of real analysis.

Keywords

Borelexceptional setnondecreasing functionsNevanlinna

MSC classification

Secondary: 26A48: Monotonic functions, generalizations 30D35: Distribution of values, Nevanlinna theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 3 , June 2010 , pp. 353 - 361
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The first author’s research was partially supported by an EPSRC Advanced Research Fellowship and a project grant from the Leverhulme Trust. The second author’s research was partially supported by the Academy of Finland (grant no. 118314).

References

[1]Borel, E., ‘Sur les zéros des fonctions entières’, Acta Math. 20 (1897), 357–396.CrossRefGoogle Scholar
[2]Cherry, W. and Ye, Z., Nevanlinna’s Theory of Value Distribution (Springer, Berlin, 2001).CrossRefGoogle Scholar
[3]Hayman, W. K., Meromorphic Functions (Clarendon Press, Oxford, 1964).Google Scholar
[4]Hayman, W. K., ‘On the characteristic of functions meromorphic in the plane and of their integrals’, Proc. London Math. Soc. 14a (1965), 93–128.CrossRefGoogle Scholar
[5]Hinkkanen, A., ‘A sharp form of Nevanlinna’s second fundamental theorem’, Invent. Math. 108 (1992), 549–574.CrossRefGoogle Scholar
[6]Nevanlinna, R., ‘Zur Theorie der meromorphen Funktionen’, Acta Math. 46 (1925), 1–99.CrossRefGoogle Scholar
[7]Nevanlinna, R., ‘Remarques sur les fonctions monotones’, Bull. Sci. Math., II. Sér. 55 (1931), 140–144.Google Scholar
[8]Weyl, H., Meromorphic Functions and Analytic Curves, Annals of Mathematics Studies, 12 (Princeton University Press, Princeton, NJ, 1943).Google Scholar