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 $n\ge2$, $0\le l\le n-2$, $-1\le m\le l$, $m+l\le n-3$, $1\le p,q,r\le\infty$. 
@article{MZM_1997_62_5_a8,
     author = {A. I. Zvyagintsev},
     title = {Strict inequalities for the derivatives of functions satisfying certain boundary conditions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {712--724},
     publisher = {mathdoc},
     volume = {62},
     number = {5},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_5_a8/}
}
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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/