On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives

Abstract

New sharp inequalities of the Kolmogorov type are established, in particular, the following sharp inequality for $2 \pi$-periodic functions $x \in L^r_{\infty}(T):$ $$||x^{(k)}||_l \leq \left(\frac{\nu(x')}{2} \right)^{\left(1 - \frac1p \right)\alpha} \frac{||\varphi_{r-k}||_l}{||\varphi_r||^{\alpha}_p} ||x||^{\alpha}_p \left|\left|x^{(r)}\right|\right|^{1-\alpha}_{\infty},$$ $k,\;r \in \mathbb{N},\quad k < r, \quad r \geq 3,\quad p \in [1, \infty],\quad \alpha = (r-k) / (r - 1 + 1/p), \quad \varphi_r$ is the perfect Euler spline of order $r,\quad \nu(x')$ is the number of sign changes of the derivative $x'$ on a period.