Multiplicative inequalities for derivatives, and a priori estimates of smoothness of solutions of nonlinear differential equations

Inequalities of the following form are proved: if $x\in C^n[a,b]$ is an arbitrary function and $r=(\alpha_1\cdot1+\dots+\alpha_n\cdot n)/(\alpha_0+\dots+\alpha_n)$, then
$$ \|x^{(r)}\|_C\leqslant c\bigl\||x|^{\alpha_0}|x'|^{\alpha_1}\cdot\ldots\cdot|x^{(n)}|^{\alpha_n}\bigr\|_C, $$
where $c$ depends only on $\alpha_0,\dots,\alpha_n$. The exponent $r$ is a limiting exponent. With the inequalities as a basis, imbedding theorems are constructed for classes of solutions of nonlinear singular differential equations in the space of $r$ times differentiable functions.