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Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis

Authors:

Institution:

(1) Dnepropetrovsk National University, Dnepropetrovsk;(2) Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk;(3) Dnepropetrovsk National Transport University, Dnepropetrovsk

Abstract:

We obtain new exact inequalities of the form

$\left\| {x^{(k)} } \right\|_q \leqslant K\left\| x \right\|_p^\alpha \left\| {x^{(r)} } \right\|_s^{1 - \alpha }$

for functions defined on the axis R or the semiaxis R ₊ in the case where

$r = 2, k = 0, p \in (0,\infty ), q \in (0,\infty ], q > p, s = 1,$

for functions defined on the axis R in the case where

$r = 2, k = 1, q \in 2,\infty ), p = \infty , s = 1,$

and for functions of constant sign on R or R ₊ in the case where

$r = 2, k = 0, p \in (0,\infty ), q \in (0,\infty ], q > p, s = \infty$

and in the case where

$r = 2, k = 1, p \in (0,\infty ), q = s s = \infty$

. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 291–302, March, 2006.

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