Geodesic
Inequalities between the norms of a function and its derivatives
Eurasian mathematical journal, Tome 7 (2016) no. 1, pp. 28-49.

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The paper is devoted to the problem of finding the maximum of the norm $||x||_q$ with the constraints $||x||_p=\eta$, $||\dot{x}||_r=\sigma$, $x(0)=a$, $a, \sigma, \eta>0$, for functions $x\in L_p(\mathbb{R}_-)$ with derivatives $\dot{x}\in L_r(\mathbb{R_-})$, $0 p \leqslant q \infty$, $r > 1$. The arguments employed are based on the standard machinery of the calculus of variations. 
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A. S. Kochurov. Inequalities between the norms of a function and its derivatives. Eurasian mathematical journal, Tome 7 (2016) no. 1, pp. 28-49. http://geodesic.mathdoc.fr/item/EMJ_2016_7_1_a2/

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