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

We obtain new sharp Kolmogorov-type inequalities, in particular the following sharp inequality for 2π-periodic functions x ∈ L ^r_∞ (T):

$$ {\left\| {{x^{(k)}}} \right\|_1} \leq {\left( {\frac{{v\left( {x'} \right)}}{2}} \right)^{\left( {1 - \frac{1}{p}} \right)\upalpha }}\frac{{{{\left\| {{\upvarphi_{r - k}}} \right\|}_1}}}{{\left\| {{\upvarphi_r}} \right\|_p^\upalpha }}\left\| x \right\|_p^\upalpha \left\| {{x^{(r)}}} \right\|_\infty^{1 - \upalpha }, $$

where k, r ∈ N, k < r, r ≥ 3, p ∈ [1, ∞], α = (r – k) / (r – 1 + 1/p), φ _r is the perfect Euler spline of order r, and ν(x′) is the number of sign changes of x′ on a period.