Geodesic
On Sharp Constants in Inequalities for the Modulus of a~Derivative
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 104-126

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For every $1\le r\le\infty$, we solve a Kolmogorov-type problem of describing all triples of numbers $\mu _0,\mu _1,\mu _2\ge 0$ for which there exists a function $f$ with an absolutely continuous derivative on the interval $[0,1]$ such that $\|f\|_{L_\infty (0,1)}=\mu _0$, $|f'(x)|=\mu _1$, and $\|f''\|_{L_r(0,1)}=\mu _2$, where $x$ is a fixed point in the interval $[0,1]$. 
@article{TM_2003_243_a9,
     author = {V. I. Burenkov and V. A. Gusakov},
     title = {On {Sharp} {Constants} in {Inequalities} for the {Modulus} of {a~Derivative}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {104--126},
     publisher = {mathdoc},
     volume = {243},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2003_243_a9/}
}
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V. I. Burenkov; V. A. Gusakov. On Sharp Constants in Inequalities for the Modulus of a~Derivative. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 104-126. http://geodesic.mathdoc.fr/item/TM_2003_243_a9/