@article{Kofanov_Popovich_2020, title={Sharp Remez type inequalities of various metrics with non-symmetric restrictions on functions}, volume={72}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/2352}, DOI={10.37863/umzh.v72i7.2352}, abstractNote={<p>UDC 517.5</p> <p>For any $p\in (0, \infty],$ $\omega &gt; 0,$ $\beta \in (0, 2 \omega)$, and arbitrary measurable set $B \subset I_d := [0, d],$ $\mu B \le \beta,$ we obtain the sharp inequality of Remez type<br>$$<br>\|x_{\pm}\|_\infty \le<br>\frac{\|(\varphi+c)_{\pm}\|_\infty}{\|\varphi+c\|_{L_p(I_{2\omega}<br>\setminus B^c_y) } \left\|x \right\|_{L_{p} \left(I_d \setminus B<br>\right)}<br>$$<br>on the set $S_{\varphi}(\omega)$ of $d$-periodic functions $x$ having zeros with given the sine-shaped $2\omega$-periodic comparison function $\varphi$, where $c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$ satisfies the condition<br>$$<br>\|x_{+}\|_\infty \cdot<br>\|x_{-}\|^{-1}_\infty = \|(\varphi+c)_{+}\|_\infty \cdot<br>\|(\varphi+c)_{-}\|^{-1}_\infty ,<br>$$<br>$B^c_y:=\{t\in [0, 2\omega]:|\varphi(t)+c| &gt; y \}$ and $y$ is such that $\mu B^c_y = \beta$.</p> <p>In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and splines with given quotient $\|x_{+}\|_\infty / \|x_-\|_\infty$.</p&gt;}, number={7}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Kofanov, V. A. and Popovich, I. V.}, year={2020}, month={Jul.}, pages={918-927} }