SUMMARY
For odd r?Nr\in \mathbb{N}; a,ß>0\alpha, \beta >0; p?[1,8]p\in [1, \infty]; d?(0,2p)\delta \in (0, 2 \pi), any 2p2\pi-periodic function x?Lr8(I2p)x\in L^r_{\infty}(I_{2\pi}), I2p:=[0,2p]I_{2\pi}:=[0, 2\pi], and arbitrary measurable set B?I2p,B \subset I_{2\pi}, µB?d/?,\mu B \leqslant \delta/\lambda, where ?=\lambda= (?fa,ßr?8?a-1x(r)++ß-1x(r)-?8E-10(x)8)1/r\left({\left\|\varphi_{r}^{\alpha, \beta}\right\|_{\infty} \left\| {\alpha^{-1}}{x_+^{(r)}} + {\beta^{-1}}{x_-^{(r)}}\right\|_\infty}{E^{-1}_0(x)_\infty}\right)^{1/r}, we obtain sharp Remez type inequality E0(x)8??fa,ßr?8E0(fa,ßr)?Lp(I2p\Bd)?x??Lp(I2p\B)?a-1x(r)++ß-1x(r)-?1-?8,E_0(x)_\infty \leqslant \frac{\|\varphi_r^{\alpha, \beta}\|_\infty}{E_0(\varphi_r^{\alpha, \beta})^{\gamma}_{L_p(I_{2\pi} \setminus B_\delta)}} \left\|x \right\|^{\gamma}_{{L_p} \left(I_{2\pi} \setminus B \right)}\left\| {\alpha^{-1}}{x_+^{(r)}} + {\beta^{-1}}{x_-^{(r)}}\right\|_\infty^{1-\gamma}, where ?=rr+1/p,\gamma=\frac{r}{r+1/p}, fa,ßr\varphi_r^{\alpha, \beta} is non-symmetric ideal Euler spline of order rr, Bd:=[M-d2,M+d1]B_\delta:= \left[M- \delta_2, M+ \delta_1 \right], MM is the point of local maximum of spline fa,ßr\varphi_r^{\alpha, \beta} and d1>0\delta_1 > 0, d2>0\delta_2 > 0 are such that fa,ßr(M+d1)=fa,ßr(M-d2),d1+d2=d.\varphi_r^{\alpha, \beta}(M+ \delta_1) = \varphi_r^{\alpha, \beta}(M- \delta_2), \;\; \delta_1 + \delta_2 = \delta .In particular, we prove the sharp inequality of Hörmander-Remez type for the norms of intermediate derivatives of the functions x?Lr8(I2p)x\in L^r_{\infty}(I_{2\pi}).