Ukrainian Mathematical Journal

, Volume 71, Issue 6, pp 896–911

# Bojanov–Naidenov Problem for Differentiable Functions on the Real Line and the Inequalities of Various Metrics

• V. A. Kofanov
Article

For given r ∈ N, p, λ > 0 and any fixed interval [a, b] R, we solve the extreme problem

$$\underset{a}{\overset{b}{\int }}{\left|x(t)\right|}^q dt\to \sup, \kern0.5em q\ge p,$$
on a set of functions $$x\in {L}_{\infty}^r$$ such that
$${\displaystyle \begin{array}{ccc}{\left\Vert {x}^{(r)}\right\Vert}_{\infty}\le 1,& {\left\Vert x\right\Vert}_{p,\updelta}\le {\left\Vert {\upvarphi \uplambda}_{,r}\right\Vert}_{p,\updelta, }& \delta \in \Big(0,\uppi /\uplambda \end{array}},$$

where

$${\displaystyle \begin{array}{cc}{\left\Vert x\right\Vert}_{p,\delta}:= \sup \left\{\left\Vert x\right\Vert {L}_{p\left[a,b\right]}:a,b\in \mathbf{R},\right.& \left.0<b-a\le \updelta \right\}\end{array}}$$

and φ⋌,r is a (2𝜋/λ)-periodic Euler spline of order r. In particular, we solve the same problem for the intermediate derivatives x(k), k=1, . . . , r−1, with q≥1. In addition, we prove the inequalities of various metrics for the quantities ||x||p,𝜹.

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