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

Article
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 −βx(r)(t) ≤ 𝛼 for almost all tR, 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 {‖xLp[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.

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