Sharp inequalities of various metrics on the classes of functions with given comparison function

Abstract

For any $$$q > p > 0$$$, $$$\omega > 0,$$$ $$$d \ge 2 \omega,$$$  we obtain the following sharp inequality of various metrics
$$
\|x\|_{L_q(I_{d})} \le \frac{\|\varphi +
c\|_{L_q(I_{2\omega})}}{\|\varphi + c \|_{L_p(I_{2\omega})}}
\|x\|_{L_p(I_{d})}
$$
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 $$$c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$$$ is such that
$$
\|x_{\pm}\|_{L_p(I_{d})} = \|(\varphi +
c)_{\pm}\|_{L_p(I_{2\omega})}\,.
$$
In particular, we  obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms $$$\|x_{+}\|_{L_p(I_{d})} / \|x_-\|_{L_p(I_{d})}$$$.