  In the one-parameter family of power-law maps of the form
$f_a(x)=-|x|^{\alpha}+a,$ $\alpha >1,$ we give examples of mutually related
dynamically determined quantities, depending on the parameter $a$, such that
one is a Pick function of the following one. These Pick functions are
extendable by reflection through the $(1,+\infty)$ half-axis and have
completely monotone derivatives there.
