Let $\phi$ be the $C^\infty$ function defined on $\mathbb R_+^*$ by $e^{-t^{-2}}$ and by $0$ on $\mathbb R_-$.

Question: I think that there exists $\rho>0$ such that $$ \forall t\in \mathbb R,\forall n\in \mathbb N,\quad\vert{\phi^{(n)}(t)}\vert\le (n!)^{3/2} \rho^{1+n}, \tag{$\ast$} $$ but I do not have a simple proof. I would like to know if an elementary argument could provide the above estimates.

As an interesting byproduct of this global Gevrey estimate of order $3/2$, there is this nice counter-example by A.N. Tychonov violating Cauchy uniqueness for the heat equation with $$ u(x,t)=\sum_{n\ge 0}\phi^{(n)}(t)\frac{x^{2n}}{(2n)!},\quad \partial_t u-\partial_x^2 u=0, \quad u_{\vert t\le 0}=0. $$ Of course to prove convergence in $C^\infty$ of the series defining $u$, some estimates are needed and $(\ast)$ is sufficient.