Value distribution and linear operators

TL;DR

This paper extends classical value distribution results like Nevanlinna's and Picard's theorems to meromorphic functions under general linear operators, including differential and difference operators.

Contribution

It introduces a framework for value distribution theory involving linear operators beyond derivatives, generalizing key theorems in complex analysis.

Findings

Generalized Nevanlinna's second main theorem for linear operators

Derived Picard's theorem for functions with linear operator derivatives

Established defect relations for these generalized settings

Abstract

Nevanlinna's second main theorem is a far-reaching generalisation 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. In this paper 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 generalisations of Picard's Theorem and the defect relations.