Bojanov – Naidenov problem for the differentiable functions on the real line and the inequalities of various metrics

Abstract

UDC 517.5
For given $r \in {\rm \bf N},$ $p,\lambda > 0$ and any fixed interval $[a, b]\subset {\rm \bf R}$ we solve the extremal problem $$ \int\limits_{a}^{b} |x(t)|^q dt \to \sup, \quad q\ge p, $$ on a set of functions $x\in L^r_{\infty}$ such that $$ \|x^{(r)}\|_{\infty} \le 1,\quad \|x\|_{p, \delta} \le \|\varphi_{\lambda, r}\|_{p, \delta}, \quad \delta \in (0, \pi/ \lambda], $$ where $$ \|x\|_{p, \delta}:=\sup \{ \|x\|_{L_p[a,\, b]}\colon a, \,b \in {\rm \bf R}, \; 0< b-a \le \delta \} $$ and $\varphi_{\lambda, r}$ is the $(2\pi/\lambda)$-periodic Euler spline of order $r.$ In particular, we solve the same problem for the intermediate derivatives $x^{(k)},$ $k=1,\ldots,r-1,$ with $q \ge 1.$ In addition, we prove the inequalities of various metrics for the quantities $\|x\|_{p, \delta}.$