One-parameter families of functions in the Pick class

Abstract

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.