Completely monotonic and related functions (Q1353643)

Let \(L^N\) denote the class of functions defined by \[ f\in L^N\Leftrightarrow (-1)^k f^{(k)}(t)\geq 0,\quad \forall t>0,\quad \forall k,\quad 0\leq k\leq N. \] For \(N\to \infty\) we write \(f\in L\); such functions are well known as completely monotonic on \((0,\infty)\). The implication \[ f\in L^N\Rightarrow [\forall \alpha> 1: f^\alpha\in L^N] \] is true for \(0\leq N\leq 5\), but false for \(N\geq 6\), while the implication \[ f\in L\Rightarrow [\forall \alpha>1: f^\alpha\in L^N] \] is true for \(0\leq N\leq 6\), but false for \(N\geq 20\).