  We study the rate of growth of ratios of intervals delimited by the
post-critical orbit of a map in the quasi-quadratic family $x\mapsto
-|x|^\alpha +a.$ The critical order $\alpha$ is an arbitrary real number
$\alpha>1.$ The range of the parameter $a$ is confined to an interval
$(1,a_{\alpha})$ of length depending on the critical order. We prove that in
every power-law family there is a unique parameter $p_{\alpha}$ corresponding
to the kneading sequence $RLRRRLRC.$ Subsequently, we obtain monotonicity
results concerning ratios of all intervals labeled by infinite post-critical
orbit in the case of the kneading sequence $RLRL...$ This extends the results
from \cite{P}, via refinement of the tools based on special properties of
power-law mappings in non-euclidean metric.
