The Growth Irregularity of Slowly Growing Entire Functions

We show that entire transcendental functions $f$ satisfying
$$ \log M(r,f)=o(\log^2r),\qquad r\to\infty\quad (M(r,f):=\max_{|z|=r}|f(z)|) $$
necessarily have growth irregularity, which increases as the growth diminishes. In particular, if $1<p<2$, then the asymptotics
$$ \log M(r,f)=\log^pr+o(\log^{2-p}r),\qquad r\to\infty, $$
is impossible. It becomes possible if "$o$" is replaced by "$O$."