Geodesic
Strict inequalities for the derivatives of functions satisfying certain boundary conditions
Matematičeskie zametki, Tome 62 (1997) no. 5, pp. 712-724.

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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 $n\ge2$, $0\le l\le n-2$, $-1\le m\le l$, $m+l\le n-3$, $1\le p,q,r\le\infty$. 
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A. I. Zvyagintsev. Strict inequalities for the derivatives of functions satisfying certain boundary conditions. Matematičeskie zametki, Tome 62 (1997) no. 5, pp. 712-724. http://geodesic.mathdoc.fr/item/MZM_1997_62_5_a8/

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