Bojanov – Naidenov problem for the differentiable functions on the real line and the inequalities of various metrics - Kofanov, V. A. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
UDC 517.5 For given $r \in {\rm \bf N},$ $p,\lambda > 0$ and any fixedinterval $[a, b]\subset {\rm \bf R}$ 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 splineof order $r.$ In particular, we solve the same problem forthe 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$\|x\|_{p, \delta}.$