  It is shown that the order and the lower order of growth are equal for all
non-trivial solutions of $f^{(k)}+A f=0$ if and only if the coefficient $A$ is
analytic in the unit disc and $\log^+ M(r,A)/\log(1-r)$ tends to a finite limit
as $r\to 1^-$. A family of concrete examples is constructed, where the order of
solutions remain the same while the lower order may vary on a certain interval
depending on the irregular growth of the coefficient. These coefficients emerge
as the logarithm of their modulus approximates smooth radial subharmonic
functions of prescribed irregular growth on a sufficiently large subset of the
unit disc. A result describing the phenomenon behind these highly non-trivial
examples is also established. En route to results of general nature, a new
sharp logarithmic derivative estimate involving the lower order of growth is
discovered. In addition to these estimates, arguments used are based, in
particular, on the Wiman-Valiron theory adapted for the lower order, and on a
good understanding of the right-derivative of the logarithm of the maximum
modulus.
