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): 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. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1642–1649, December, 2008.