Boyanov–Naydenov problem and the inequalities of various metrics

• Datum: 2026
• Hauptverfasser: Doroshenko, D., Kofanov, V., Дорошенко, Данііл, Кофанов, Володимир, Кофанов, Владимир Александрович
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2026
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/9328
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.51 We solve the Boyanov–Naidenov problem $\|x\|_{q, \delta} \to \sup$ on classes of differentiable functions $ W^r_{p, \omega}(A_0, A_r) := \{x\in L^r_{\infty}\colon \|x\|_{p, \omega} \le A_0,$ $ \|x^{(r)}\|_{\infty} \le A_r \}, $ where $\|x\|_{p, \, \delta} := \sup \{\|x\|_{L_p[a, b]}\colon  a, b \in {\rm \bf R},$ $ 0< b-a \le \delta \}; \; p, \delta > 0; $ $q \ge p;$ $\omega := \pi / \lambda,$ the number $\lambda$ satisfies the condition $ A_0 = A_r \|\varphi_{\lambda, r}\|_{p, \, \pi / \lambda}, $   $\varphi_{\lambda, r}(t) := \lambda^{-r}\varphi_{r}(\lambda t),$ and $\varphi_{r}$ is the ideal Euler spline of order $r.$ In addition, we prove that the Boyanov–Naidenov problem is equivalent to the problem of finding the sharp constant $C = C(\lambda)$ in the inequality of different metrics\begin{align}\|x\|_{q, \delta} \leq C \|x\|_{p, \omega}^{\alpha} \big\|x^{(r)}\big\|_\infty^{1\alpha},\quad x\in L^{r,\lambda}_{p,\varepsilon},\tag{1}\end{align}where $\alpha = \dfrac{r + 1/q}{r + 1/p},$ $L^{r, \lambda}_{p, \varepsilon} := \{x\in L^r_{\infty}\colon \|x\|_{p, \varepsilon} = \|\varphi_{\lambda, r}\|_{p, \varepsilon}\|x^{(r)}\|_{\infty} \},$ and $\lambda > 0.$ In particular, we obtain inequalities of the form (1) sharp in the classes $L^{r, \lambda}_{p, \varepsilon}.$ We also solve the Boyanov–Naidenov problem in the spaces of trigonometric polynomials and splines and establish theorems on the relationship between this problem and sharp inequalities of different metrics. As a consequence, sharp inequalities of this kind are proved for polynomials and splines.