TY - JOUR AU - V. A. Kofanov PY - 2009/06/25 Y2 - 2024/03/28 TI - On some extremal problems of different metrics for differentiable functions on the axis JF - Ukrains’kyi Matematychnyi Zhurnal JA - Ukr. Mat. Zhurn. VL - 61 IS - 6 SE - Research articles DO - UR - https://umj.imath.kiev.ua/index.php/umj/article/view/3057 AB - For an arbitrary fixed segment $[α, β] ⊂ R$ and given $r ∈ N, A_r, A_0$, and $p > 0$, we solve the extremal problem $$∫^{β}_{α} \left|x^{(k)}(t)\right|^qdt → \sup,\; q⩾p,\; k=0,\; q⩾1,\; 1 ⩽ k ⩽ r−1,$$on the set of all functions $x ∈ L^r_{∞}$ such that $∥x (r)∥_{∞} ≤ A_r$ and $L(x)_p ≤ A_0$, where$$L(x)p := \left\{\left( ∫^b_a |x(t)|^p dt\right)^{1/p} : a,b ∈ R,\; |x(t)| > 0,\; t ∈ (a,b)\right\}$$In the case where $p = ∞$ and $k ≥ 1$, this problem was solved earlier by Bojanov and Naidenov. ER -