On some extremal problems of different metrics for differentiable functions on the axis

For an arbitrary fixed segment [α, β] ⊂ R and given r ∈ N, A _r , A ₀, and p > 0, we solve the extremal problem

$$ \int\limits_\alpha^\beta {{{\left| {{x^{(k)}}(t)} \right|}^q}dt \to \sup, \,\,\,\,q \geqslant p,\,\,\,k = 0,\,\,\,q \geqslant 1,\,\,\,\,1 \leqslant k \leqslant r - 1,} $$

on the set of all functions x∈ L ^r _∞ such that ∥x ^(r)∥_∞ ≤ A _r and L(x) p ≤ A ₀, where

$$ L{(x)_p}: = \left\{ {{{\left( {\int\limits_a^b {{{\left| {x(t)} \right|}^p}dt} } \right)}^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}}:\,a,b \in R,\,\left| {x(t)} \right| > 0,\,t \in \left( {a,\,b} \right)} \right\} $$

In the case where p = ∞ and k ≥ 1, this problem was solved earlier by Bojanov and Naidenov.