Bojanov–Naidenov Problem for Functions with Asymmetric Restrictions for the Higher Derivative

For given r ∈ N, p, 𝛼, β,μ > 0, we solve the extreme problems

$$ \underset{a}{\overset{b}{\int }}{x}_{\pm}^q(t) dt\to \sup, \kern0.5em q\ge p, $$

in the set of pairs (x, I) of functions $x\in {L}_{\infty}^r$ and intervals I = [a, b] ⊂R satisfying the inequalities −β ≤ x^(r)(t) ≤ 𝛼 for almost all t ∈ R, the conditions$L{\left({x}_{\pm}\right)}_p\le L{\left({\left({\varphi}_{\leftthreetimes, r}^{\alpha, \beta}\right)}_{\pm}\right)}_p$, and the corresponding condition μ(supp_[a, b]x+) ≤ μ or μ(supp_[a, b]x−) ≤ μ, where L(x)p; = sup {‖x‖L_p[a, b]; a, b ∈ R,  |x(t)| > 0, t ∈ (a, b)}, supp_[a, b]x ±  ≔ {t ∈ [a, b] : x_±(t) > 0},, and ${\varphi}_{\leftthreetimes, r}^{\alpha, \beta }$ is an asymmetric (2π/⋋) -periodic Euler spline of order r. As a consequence, we solve the same extreme problems for the intermediate derivatives ${x}_{\pm}^{(k)}$, k = 1, . . . , r − 1, with q ≥ 1.