Expanding on a previous post I made recently, let
$$
\mathscr{P}:= \{ p(x) \in \mathbb{R}[x] \mid p(x) \ge 0,~\forall x\ge 0\}.
$$
The Pòlya-Szegö theorem (see Theorem 3.21 here) asserts that $p \in \mathscr{P}$ if and only if
$$
p(x) = f(x) + xg(x),
$$
in which $f$ and $g$ are *polynomial sum-of-squares*.

Is anything known about polynomials $p$ such that $p,~p' \in \mathscr{P}$? Obviously, these polynomials must enjoy many desirable properties (for instance, $p$ must be monotonically increasing on $[0,\infty)$).

Is it possible that if $p$, $p' \in \mathscr{P}$, then $p^{(k)} \in \mathscr{P}$ for $2 \le k \le \deg{p}-1$?

In particular, I am interested in proving that $|p(z)| \le p(|z|)$ for every $z \in \mathbb{C}$ or knowing whether this property is independent of a polynomial and its derivative belonging to $\mathscr{P}$.

Any references/ideas/partial results are greatly appreciated.