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Value Distribution and Linear Operators

Published online by Cambridge University Press:  22 November 2013

R. Halburd  and
R. Korhonen
R. Halburd
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK, (r.halburd@ucl.ac.uk)
R. Korhonen
Affiliation:
Department of Physics and Mathematics, University of Eastern Finland, PO Box 111, 80101 Joensuu, Finland, (risto.korhonen@uef.fi)
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Abstract

Nevanlinna's second main theorem is a far-reaching generalization of Picard's theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f′ appear. We show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalizations of Picard's theorem and the defect relations.

Keywords

Nevanlinnadifferential differenceramification termPicard's theoremdefect relationslinear operatorPrimary 30D35Secondary 34M0334M0539A06
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 2 , June 2014 , pp. 493 - 504
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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