  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.
