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

òab | x(k)(t) |qdt ? sup,     q 3 p,    k = 0    \textor    q 3 1,    k 3 1, \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},
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