Kolmogorov-type inequalities for functions with asymmetric restrictions on the highest derivative

Abstract

For $k, r\in {\rm \bf N}$, $k0$; $\alpha, \beta>0$ and for functions $x\in L_{\infty}^r({\rm\bf R})$ inequalities that estimate the norm $\|x_{\pm }^{(k)}\|_{L_q[a,b]}$ on an arbitrary segment $[a,b] \subset {\rm\bf R}$ such that $\;x^{(k)}(a)=x^{(k)}(b)=0$ via a local norm of the function $|||x^{\uparrow \downarrow}|||_p :=\sup \left\{ E_0(x)_{L_p[a,b]}: \; \pm x'(t) > 0 \; \forall t\in (a,b), \;\; a,b\in \rm \bf R \right\},$ and the asymmetric norm $\|\alpha^{-1}x_+^{(r)}+\beta ^{-1}x_-^{(r)}\| _{\infty}$ of its highest derivative are proved, where $E_0(x)_{L_p([a,b])}:= \inf \{\|x - c\|_{L_p([a,b])}: c \in {\rm \bf R }\}$. As a consequence, generalizations of a number of well-known Kolmogorov-type inequalities are obtained.

Keywords

• sharp kolmogorov-type inequality
• asymmetric case
• local norm