Finding functions $f(t)$ such that $(f(t))^{s}$ is a Bernstein function.

I'm studying Bernstein function from the book R.L. Schilling, R. Song, and Z. Vondraček, Bernstein Functions: Theory and Applications, 2. ed (De Gruyter, Berlin, 2012).

Definition. A function $f:(0,\infty)\to\mathbb{R}$ is a Bernstein function if $f$ is of class $C^{\infty}$,', $f(t)\geq 0$ for all $t>0$ and\begin{align}(-1)^{n-1}f^{(n)}(t)\geq 0,\text{ for all }n\in\mathbb{N}\text{ and } t>0\end{align}

An example of a Bernstein function is $f(t)=t^s$ when $0\leq s\leq 1$. In general, is there a criterion for functions $f(t)$ such that $(f(t))^{s}$ with $0\leq s\leq 1$ let a Berstein function?. Thanks.