Strict inequalities for the derivatives of functions satisfying certain boundary conditions

For functions satisfying the boundary conditions
$$ f(0)=f'(0)=\dots=f^{(m)}(0)=0,\qquad f(1)=f'(1)=\dots=f^{(l)}(1)=0, $$
the following inequality with sharp constants in additive form is proved:
$$ \|f^{(n-1)}\|_{L_q(0,1)} \le A\|f\|_{L_p(0,1)}+B\|f^{(n)}\|_{L_r(0,1)}, $$
where $n\ge2$, $0\le l\le n-2$, $-1\le m\le l$, $m+l\le n-3$, $1\le p,q,r\le\infty$.