Exact kolmogorov-type inequalities with bounded leading derivative in the case of low smoothness

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 ∈ L∞r : (Figure Presented) where α= min { l-k / r, r-k+1/q / r + 1/p} and (φ is the perfect Euler spline of order r. © 2001 Plenum Publishing Corporation.