2013/09/12 by Halburd, Rodney, Korhonen, Risto
#34M05 #39A06 #Complex Variables (math.CV) #FOS: Mathematics #Primary 30D35 #Secondary 34M03
paper · doi:10.48550/arxiv.1309.3333
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.