Strict inequalities for the derivatives of functions satisfying certain boundary conditions (Q1277581)

For a function of a single variable defined on a finite interval and which has an absolutely continuous derivative \(f^{(n-1)}\) on this interval and which satisfy the boundary conditions \[ (1)\quad f(0)= f'(0)=\cdots= f^{(m)} (0)=0, \qquad (2)\quad f(1)= f'(1)=\cdots= f^{(l)}(1)= 0, \] the following inequality with sharp constants in additive form is proved: \[ (3)\quad \| f^{(n-1)} \|_{L_q(0,1)}\leq A\| f\|_{L_p(0,1)}+ B\| f^{(n)} \|_{L_r(0,1)}, \] where \(n\geq 2\), \(0\leq l\leq n-2\), \(-1\leq m\leq l\), \(m+l\leq n-3\), and \(1\leq p,q,r\leq \infty\) (in the case \(m=1\) we assume that condition (1) is not imposed).