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

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

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

.