### Strengthening the Comparison Theorem and Kolmogorov Inequality in the Asymmetric Case

V.A. Kofanov (Oles Honchar Dnipro National University), https://orcid.org/0000-0003-0392-2257
K.D. Sydorovych (Oles Honchar Dnipro National University)

#### Abstract

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
$$|||x|||_\infty:=\frac12 \sup_{\alpha ,\beta}\{ |x(\beta)-x(\alpha)|:x'(t)\neq 0 \;\;\forall t\in (\alpha ,\beta) \}$$
$k,r \in \mathbb{N}$, $k<r$, $\alpha, \beta > 0$, $\varphi_r( \cdot \;;\alpha ,\beta )_r$ 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.

#### Keywords

Kolmogorov comparison theorem; Kolmogorov inequality; asymmetric case; strengthening

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#### References

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Babenko V.F. "Asymmetric extremal problems in approximation theory", Dokl. AN SSSR, 1983; 269(3): pp. 521-524. (in Russian)

Kofanov V.A. "Strengthening the comparison theorem and Kolmogorov's inequality and their applications", Ukrainian Math. J., 2002; 54(10): pp. 1348-1355. doi:10.1023/A:1023728202727

Kofanov V.A. "On a strengthening of Kolmogorov's inequality", Res. Math., 2001; 6: pp. 63-67. (in Russian)

DOI: https://doi.org/10.15421/242204

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