Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness - Babenko, V. F.; Kofanov, V. A.; Pichugov, S. A. (Ukrains’kyi Matematychnyi Zhurnal, 2001)
We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p ∈ [1, ∞], the following unimprovable inequality holds for functions \(x \in L_∞ ^r \) : $$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{φ }_{r - k} } \right\|_q }}{{\left\| {{φ }_r } \right\|_p^α }}\left\| x \right\|_p^α \left\| {x^{\left( k \right)} } \right\|_∞ ^{1 - α } $$ where \(α = \min \left\{ {1 - \frac{k}{r},\frac{{r - k + {1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-0em} q}}}{{r + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0em} p}}}} \right\}\) and ϕ r is the perfect Euler spline of order r.