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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 relateddynamically determined quantities, depending on the parameter $a$, such thatone is a Pick function of the following one. These Pick functions areextendable by reflection through the $(1,+\infty)$ half-axis and havecompletely monotone derivatives there.

One-parameter families of functions in the Pick class