We solve the Bojanov–Naidenov problem \({\Vert {x}^{\left(k\right)}\Vert }_{q,\delta }\to \text{sup}, k=1,\dots ,r-1,q\ge 1,\) on the classes of functions \({W}_{p,\varepsilon }^{r}\left({A}_{0},{A}_{r}\right):=\left\{x\in {L}_{\infty }^{r}:{\Vert x\Vert }_{p,\varepsilon }\le {A}_{0}{\Vert {x}^{\left(r\right)}\Vert }_{\infty }\le {A}_{r}\right\},\) where \({\Vert x\Vert }_{p,\delta }:=\text{sup}\left\{{\Vert x\Vert }_{{L}_{p}\left[a,b\right]}:a,b\in \mathbf{R},0<b-a\le \delta \right\},p,\delta >0,\varepsilon \in \left(0,\left.{\varepsilon }_{1}\right],{\varepsilon }_{1}:=\pi /\omega ,\right.\) the number ω satisfies the condition \({A}_{0}={A}_{r}{\Vert {\varphi }_{\omega ,r}\Vert }_{p,\pi /\omega },{\varphi }_{\omega ,r}\left(t\right):={\omega }^{-r}{\varphi }_{r}\left(\omega t\right),\) and \({\varphi }_{r}\) is an ideal Euler spline of order r. In addition, we prove that the Bojanov–Naidenov problem is equivalent to the problem of sharp constant \(C=C\left(\lambda \right)\) in the Kolmogorov-type inequality 1 \({\Vert {x}^{\left(k\right)}\Vert }_{q,\delta }\le C{\Vert x\Vert }_{p,\varepsilon }^{\alpha }{\Vert {x}^{\left(r\right)}\Vert }_{\infty }^{1-\alpha },x\in {L}_{p,\varepsilon }^{r,\lambda },\)
where \(\alpha =\frac{r-k+1/q}{r+1/p},{L}_{p,\varepsilon }^{r,\lambda }:=\left\{x\in {L}_{\infty }^{r}:{\Vert x\Vert }_{p,\varepsilon }={\Vert {\varphi }_{\lambda ,r}\Vert }_{p,\varepsilon }\bullet {\Vert {x}^{\left(r\right)}\Vert }_{\infty }\right\},\) and \(\lambda >0\) . In particular, we obtain sharp inequalities of the form (1) on the classes \({L}_{p,\varepsilon }^{r,\lambda }\) .
We also solve the Bojanov–Naidenov problem in the spaces of trigonometric polynomials and splines and prove the theorems on the relationship between the analyzed problem and sharp Bernstein-type inequalities. As a consequence, we prove sharp inequalities of the indicated type for polynomials and splines.
