Let $f \in C^\infty (\Bbb R) $, and $f(0) = 0$. Assume if $|w| \leq \delta$ then $| f(w) | \leq c |w|^a$ for some fixed $a \in \Bbb N$.
Now let $\| w \|_{L^\infty(\Bbb R)} \leq \delta$. Then can we derive an estimate for $\| f (w) \|_{C^k ( [0,\delta])}$ by using the inequality $|f(w)| \leq c |w|^a$ ?
Here $\| f (w) \|_{C^k ( [0,\delta])}$ means the supremum of all the functions up to differentiation order $k$ on $[0,\delta]$.