# 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 $$|g|_{C^{0,alpha}} = sup_{x,y in (0,1)} frac{|g(x)-g(y)|}{|x-y|^alpha}$$?