Order of growth of distributional irregular entire functions for the differentiation operator

TL;DR

This paper investigates the growth rates of entire functions that are distributionally irregular for the differentiation operator, establishing existence results with controlled growth and dense submanifolds of such functions.

Contribution

It proves the existence of distributionally irregular entire functions with specific growth bounds and constructs dense submanifolds where all nonzero vectors share this growth behavior.

Findings

Existence of distributionally irregular entire functions with growth not exceeding e^r r^{-b}

Construction of dense linear submanifolds of such functions

Completes known results on growth rates of D-hypercyclic entire functions

Abstract

We study the rate of growth of entire functions that are distributionally irregular for the differentiation operator D. More specifically, given $p \in [1,\infty ]$ and $b \in (0,a)$, where $a = \frac{1}{2 max\{2,p\}}$, we prove that there exists a distributionally irregular entire function $f$ for the operator D such that its p-integral mean function $M_p(f,r)$ grows not more rapidly than $e^r r^{-b}$. This completes related known results about the possible rates of growth of such means for D-hypercyclic entire functions. It is also obtained the existence of dense linear submanifolds of H(C) all whose nonzero vectors are D-distributionally irregular and present the same kind of growth.