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We study the rate of growth of ratios of intervals delimited by thepost-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 inevery power-law family there is a unique parameter $p_{\alpha}$ correspondingto the kneading sequence $RLRRRLRC.$ Subsequently, we obtain monotonicityresults concerning ratios of all intervals labeled by infinite post-criticalorbit in the case of the kneading sequence $RLRL...$ This extends the resultsfrom \cite{P}, via refinement of the tools based on special properties ofpower-law mappings in non-euclidean metric.

Decreasing families of dynamically determined intervals in the power-law family