Strengthening the Comparison Theorem and Kolmogorov Inequality in the Asymmetric Case

We obtain the strengthened Kolmogorov comparison theorem in asymmetric case.In particular, it gives us the opportunity to obtain the following strengthened Kolmogorov inequality in the asymmetric case:$$\|x^{(k)}_{\pm }\|_{\infty}\le \frac{\|\varphi _{r-k}( \cdot \;;\alpha ,\beta )_\pm \|_{\infty }}{E_0(\varphi _r( \cdot \;;\alpha ,\beta ))^{1-k/r}_{\infty }}|||x|||^{1-k/r}_{\infty}\|\alpha^{-1}x_+^{(r)}+\beta^{-1}x_-^{(r)}\|_\infty^{k/r}$$for functions $x \in L^r_{\infty }(\mathbb{R})$, where$$|||x|||_\infty:=\frac12 \sup_{\alpha ,\beta}\{ |x(\beta)-x(\alpha)|:x'(t)\neq 0 \;\;\forallt\in (\alpha ,\beta) \}$$$k,r \in \mathbb{N}$, $k 0$, $\varphi_r( \cdot \;;\alpha ,\beta )_r$ is the asymmetric perfect spline of Euler of order $r$ and $E_0(x)_\infty $ is the best uniform approximation of the function $x$ by constants.