$y=f(x)$ is a $C^{\infty}$ function defined in $\mathbb{R}$,for any $k\in \mathbb{Z}^+$,we let $M_k=\mathop{\sup}_{x\in\mathbb{R}}|f^{k}(x)|$. $m,n \in \mathbb{Z}$ and $0\le m <n$.
If $M_m$ and $M_n$ are both bounded, then for which intergers $k$, $M_k$ is bounded ? And then unbounded ?
Thanks advance for your help.
By the Landau-Kolmogorov inequality, $M_k$ is bounded for $m\leq k\leq n.$ On the other hand it may be unbounded for all other $k,$ for example consider $f(x)=x^m+(1+x^2)^{-n/2}\sin(x^2).$