On some extremal problems of different metrics for differentiable functions on the axis - Kofanov, V. A. (Ukrains’kyi Matematychnyi Zhurnal, 2009)
For an arbitrary fixed segment [α, β] ⊂ R and given r ∈ N, A_r, A_0, and p > 0, we solve the extremal problem ∫^β_α |x^(k)(t)|^qdt → , q⩾p, k=0, q⩾1, 1 ⩽ k ⩽ r−1, on the set of all functions x ∈ L^r_∞ such that ∥x (r)∥_∞ ≤ A_r and L(x)_p ≤ A_0, where L(x)p := {( ∫^b_a |x(t)|^p dt)^1/ p : a,b ∈ R, |x(t)| > 0, t ∈ (a,b)} In the case where p = ∞ and k ≥ 1, this problem was solved earlier by Bojanov and Naidenov.