On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives - Kofanov, V. A.; Miropol'skii, V. E. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
New sharp inequalities of the Kolmogorov type are established, in particular, the following sharp inequality for 2 π-periodic functions x L^r_∞(T): ||x^(k)||_l ≤ ( u(x’)2 )^(1 - 1p )α ||φ_r-k||_l||φ_r||^α_p ||x||^α_p ||x^(r)||^1-α_∞, 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,u(x’) is the number of sign changes of the derivative x’ on a period.