Sharp Nagy type inequalities for the classes of functions with given quotient of the uniform norms of positive and negative parts of a function

Abstract

For any $$$p\in (0, \infty],$$$ $$$\omega > 0,$$$ $$$d \ge 2 \omega,$$$ we obtain the sharp inequality of Nagy type
$$
\|x_{\pm}\|_\infty \le
\frac{\|(\varphi+c)_{\pm}\|_\infty}{\|\varphi+c\|_{L_p(I_{2\omega}
)}} \left\|x \right\|_{L_{p} \left(I_d  \right)}
$$
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
$$
 \|x_{+}\|_\infty \cdot
\|x_{-}\|^{-1}_\infty = \|(\varphi+c)_{+}\|_\infty \cdot
\|(\varphi+c)_{-}\|^{-1}_\infty .
$$

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 $$$\|x_{+}\|_\infty / \|x_-\|_\infty$$$.