@article{Kofanov_2008, title={Inequalities for derivatives of functions in the spaces <i>L<sub>p</sub></i&gt;}, volume={60}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/3248}, abstractNote={The following sharp inequality for local norms of functions $x \in L^{r}_{\infty,\infty}(\textbf{R})$ is proved: $$\frac1{b-a}\int\limits_a^b|x’(t)|^qdt \leq \frac1{\pi}\int\limits_0^{\pi}|\varphi_{r-1}(t)|^q dt \left(\frac{||x||_{L_{\infty}(\textbf{R}) }{||\varphi_r||_{\infty }\right)^{\frac{r-1}rq}||x^{(r)}||^q_{\infty}r,\quad r \in \textbf{N},$$ where $\varphi_r$ is the perfect Euler spline, takes place on intervals $[a, b]$ of monotonicity of the function $x$ for $q \geq 1$ or for any $q &gt; 0$ in the cases of $r = 2$ and $r = 3.$ As a corollary, well-known A. A. Ligun’s inequality for functions $x \in L^{r}_{\infty}$ of the form $$||x^{(k)}||_q \leq \frac{||\varphi_{r-k}||_q}{||\varphi_r||_{\infty}^{1-k/r } ||x||^{1-k/r}_{\infty}||x^{(r)}||^{k/r}_{\infty},\quad k,r \in \textbf{N},\quad k < r, \quad 1 \leq q &lt; \infty,$$ is proved for $q \in [0,1)$ in the cases of $r = 2$ and $r = 3.$}, number={10}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Kofanov, V. A.}, year={2008}, month={Oct.}, pages={1338 -} }