fa.functional analysis – Functions for which $|f^{(k)}|_{C^{0,alpha}(0,1)} le Vert f Vert_{L^1(0,1)}$

Let $f in C^k(0,1)$ and assume that the $k$-th derivative is $alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) subset (0,1)$. Can we characterize (or at least find some examples of) non-constant functions $f$ as above such that
$$|f^{(k)}|_{C^{0,alpha}(0,1)} le Vert f Vert_{L^1(0,1)},$$
where $|g|_{C^{0,alpha}} = sup_{x,y in (0,1)} frac{|g(x)-g(y)|}{|x-y|^alpha}$?