Value Distribution and Linear Operators

Abstract

AbstractNevanlinna'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.