For each $n \in \mathbb{N}$, the Hermite function $\psi_n : \mathbb{R} \to \mathbb{R}$ is a Schwartz function defined by \begin{equation} \psi_n(x):=(-1)^n(2^n n!\sqrt{\pi})^{-1/2} e^{x^2/2} \frac{d^n}{dx^n}e^{-x^2} \end{equation} according to the Wikipedia article : https://en.wikipedia.org/wiki/Hermite_polynomials#Differential-operator_representation

Moreover, it is written there that each Hermite function satisfies the $L^\infty$ bound \begin{equation} \lvert \psi_n(x) \rvert \leq \pi^{-1/4} \text{ for all } x \in \mathbb{R}, n \in \mathbb{N} \end{equation} which is known as the Cramer's inequality.

Now, I am trying to find a bound of \begin{equation} \sup_{x \in \mathbb{R}} \lvert x^m \psi_n(x) \rvert \end{equation} for any given $n,m \in \mathbb{N}$.

More specifically, I am curious about the asymptotic behavior of this supremum as $n \to \infty$ while $m$ is fixed. Would it remain bounded or diverge?

I searched for references myself but cannot get an idea about resolving this issue. Could anyone please help me?