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

Abstract

For $$$k, r\in {\rm \bf N}$$$, $$$k<r$$$; $$$q\ge 1$$$, $$$p>0$$$; $$$\alpha, \beta>0$$$ 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.