Inequalities for derivatives of functions in the spaces Lp - Kofanov, V. A. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
The following sharp inequality for local norms of functions $x \in L^{r}_{∞,∞}(R)$ is proved:$$\frac1{b-a}\int\limits_a^b|x'(t)|^qdt ≤ \frac1π\int\limits_0^{π}|φ_{r-1}(t)|^q dt \left(\frac{||x||_{L_{∞}(R)}}{||φ_r||_{∞}}\right)^{\frac{r-1}rq}||x^{(r)}||^q_{∞}r,\quad r \in N,$$ where $φ_r$ is the perfect Euler spline, takes place on intervals $[a, b]$ of monotonicity of the function $x$ for $q ≥ 1$ or for any $q > 0$ in the cases of $r = 2$ and $r = 3.$As a corollary, well-known A. A. Ligun's inequality for functions $x \in L^{r}_{∞}$ of the form$$||x^{(k)}||_q ≤ \frac{||φ_{r-k}||_q}{||φ_r||_{∞}^{1-k/r}} ||x||^{1-k/r}_{∞}||x^{(r)}||^{k/r}_{∞},\quad k,r \in N,\quad k < r, \quad 1 ≤ q < ∞,$$is proved for $q \in [0,1)$ in the cases of $r = 2$ and $r = 3.$