Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

For arbitrary [α, β] ⊂ R and p > 0, we solve the extremal problem

$$ \int\limits_\alpha^\beta {{{\left| {{x^{(k)}}(t)} \right|}^q}dt \to \sup, \quad q \geq p,\quad k = 0\quad {\text{or}}\quad q \geq 1,\quad k \geq 1}, $$

on the set of functions $S_\varphi^k$ such that φ ⁽ⁱ⁾ is a comparison function for x ⁽ⁱ⁾, = 0, 1, …, k, and (in the case k = 0) L(x) _p  ≤ L(φ) _p , where

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

In particular, we solve this extremal problem on Sobolev classes and on bounded subsets of the spaces of trigonometric polynomials and splines.