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

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.