How do you prove the following estimate for composition of functions:
If $k\geq 1$, then there exists a constant $c=c(k,\alpha)$ such that $$ \|f_1\circ g_1-f_2\circ g_2\|_{k,\alpha}\leq c\left(1+\|\nabla g_1\|_{k-1,\alpha}+\|\nabla g_2\|_{k-1,\alpha}\right)^{k+1}\\ \bigg[\|f_1-f_2\|_{k,\alpha}+\min\big\{\|\nabla f_1\|_{k,\alpha},\|\nabla f_2\|_{k,\alpha}\big\} \|g_1-g_2\|_{k,\alpha}\bigg]. $$
for functions $f$ and $g$ suffiently regular for everything to make sense. Here, the norms are the Holder norms and seminorms (i.e. see http://en.wikipedia.org/wiki/H%C3%B6lder_condition). Thank you in advance.