Studies of Differences from the point of view of Nevanlinna Theory

TL;DR

This paper advances Nevanlinna Theory by establishing optimal conditions for shift invariance, developing a difference logarithmic derivative for certain subharmonic functions, and analyzing difference equations under growth constraints.

Contribution

It introduces the best known conditions for shift invariance, extends the difference logarithmic derivative to hyper-order one, and classifies difference equations based on solution growth.

Findings

Optimal condition for shift invariance of counting function

Difference logarithmic derivative for hyper-order one functions

Possible forms of difference equations with growth constraints

Abstract

This paper consists of three parts. First, we give so far the best condition under which the shift invariance of the counting function, and of the characteristic of a subharmonic function, holds. Second, a difference analogue of logarithmic derivative of a $\delta$-subharmonic function is established allowing the case of hyper-order equal to one and minimal hyper-type, which improves the condition of the hyper-order less than one. Finally, we make a careful discussion of a well-known difference equation and give out the possible forms of the equation under a growth condition for the solutions.