Polynomials $p$ such that $p$ and $p'$ preserve nonnegative numbers

Question

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.

Answer 1

The equivalent condition is $p(0)\geqslant 0$, $p'(x) \geqslant 0$ for $x\geqslant 0$.Thus this class reduces to $\mathscr {P} $ by taking antiderivative.

The inequality $|p(z)|\leqslant p(|z|)$ fails for $p(z) =z^2 - z^3 +z^4$ and $z=-1$. Also $p'''\notin \mathscr{P}$.