Additive inequalities for intermediate derivatives of differentiable mappings of Banach spaces

Suppose that $X$ and $Y$ are real Banach spaces, $U\subset X$ is an open bounded set star-shaped with respect to some point, $n,k\in\mathbb N$, $k<n$, and $M_{n,k}(U,Y)$ is the sharp constant in the Markov type inequality for derivatives of polynomial mappings. It is proved that for any $M\ge M_{n,k}(U,Y)$ there exists a constant $B>0$ such that for any function$f\in C^n(U,Y)$ the following inequality holds:
$$ |\kern -.8pt|\kern -.8pt|f^{(k)}|\kern -.8pt|\kern -.8pt|_U\le M|\kern -.8pt|\kern -.8pt|f|\kern -.8pt|\kern -.8pt|_U+B|\kern -.8pt|\kern -.8pt|f^{(n)}|\kern -.8pt|\kern -.8pt|_U. $$
The constant $M=M_{n-1,k}(U,Y)$ is best possible in the sense that $M_{n-1,k}(U,Y)=\inf M$, where $\inf$ is taken over all $M$ such that for some $B>0$ the estimate holds for all $f\in C^n(U,Y)$.