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Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle

We investigate the relationship between the constants K(R) and K(T), where \(K\left( G \right) = K_{k,r} \left( {G;q,p,s;\alpha } \right): = \mathop {\mathop {\sup }\limits_{x \in L_{p,s}^r \left( G \right)} }\limits_{x^{(r)} \ne 0} \frac{{\left\| {x^{\left( k \right)} } \right\|_{L_q \left( G \ri...

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Datum:2003
Hauptverfasser: Babenko, V. F., Kofanov, V. A., Pichugov, S. A., Бабенко, В. Ф., Кофанов, В. А., Пичугов, С. А.
Format: Artikel
Sprache:Russisch
Englisch
Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2003
Online Zugang:https://umj.imath.kiev.ua/index.php/umj/article/view/3934
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Назва журналу:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
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author Babenko, V. F.
Kofanov, V. A.
Pichugov, S. A.
Бабенко, В. Ф.
Кофанов, В. А.
Пичугов, С. А.
Бабенко, В. Ф.
Кофанов, В. А.
Пичугов, С. А.
author_facet Babenko, V. F.
Kofanov, V. A.
Pichugov, S. A.
Бабенко, В. Ф.
Кофанов, В. А.
Пичугов, С. А.
Бабенко, В. Ф.
Кофанов, В. А.
Пичугов, С. А.
author_sort Babenko, V. F.
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
collection OJS
datestamp_date 2020-03-18T20:15:55Z
description We investigate the relationship between the constants K(R) and K(T), where \(K\left( G \right) = K_{k,r} \left( {G;q,p,s;\alpha } \right): = \mathop {\mathop {\sup }\limits_{x \in L_{p,s}^r \left( G \right)} }\limits_{x^{(r)} \ne 0} \frac{{\left\| {x^{\left( k \right)} } \right\|_{L_q \left( G \right)} }}{{\left\| x \right\|_{L_q \left( G \right)}^\alpha \left\| {x^{\left( r \right)} } \right\|_{L_s \left( G \right)}^{1 - \alpha } }}\) is the exact constant in the Kolmogorov inequality, R is the real axis, T is a unit circle, $$L_{p,s}^r (G)$$ is the set of functions x ∈ L p(G) such that x (r) ∈ L s(G), q, p, s ∈ [1, ∞], k, r ∈ N, k < r, We prove that if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} = 1 - k/r$$ thenK(R) = K(T),but if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} < 1 - k/r$$ thenK(R) ≤ K(T); moreover, the last inequality can be an equality as well as a strict inequality. As a corollary, we obtain new exact Kolmogorov-type inequalities on the real axis.
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spelling umjimathkievua-article-39342020-03-18T20:15:55Z Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle Сравнение точных констант в неравенствах для производных функций, заданных на вещественной оси и окружности Babenko, V. F. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. We investigate the relationship between the constants K(R) and K(T), where \(K\left( G \right) = K_{k,r} \left( {G;q,p,s;\alpha } \right): = \mathop {\mathop {\sup }\limits_{x \in L_{p,s}^r \left( G \right)} }\limits_{x^{(r)} \ne 0} \frac{{\left\| {x^{\left( k \right)} } \right\|_{L_q \left( G \right)} }}{{\left\| x \right\|_{L_q \left( G \right)}^\alpha \left\| {x^{\left( r \right)} } \right\|_{L_s \left( G \right)}^{1 - \alpha } }}\) is the exact constant in the Kolmogorov inequality, R is the real axis, T is a unit circle, $$L_{p,s}^r (G)$$ is the set of functions x ∈ L p(G) such that x (r) ∈ L s(G), q, p, s ∈ [1, ∞], k, r ∈ N, k < r, We prove that if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} = 1 - k/r$$ thenK(R) = K(T),but if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} < 1 - k/r$$ thenK(R) ≤ K(T); moreover, the last inequality can be an equality as well as a strict inequality. As a corollary, we obtain new exact Kolmogorov-type inequalities on the real axis. Досліджується взаємозв'язок між константами $K(R)$ і $K(T)$, де $$K\left( G \right) = K_{k,r} \left( {G;q,p,s;\alpha } \right): = \mathop {\mathop {\sup }\limits_{x \in L_{p,s}^r \left( G \right)} }\limits_{x^{(r)} \ne 0} \frac{{\left\| {x^{\left( k \right)} } \right\|_{L_q \left( G \right)} }}{{\left\| x \right\|_{L_q \left( G \right)}^\alpha \left\| {x^{\left( r \right)} } \right\|_{L_s \left( G \right)}^{1 - \alpha } }}$$ —точна константа в нерівності Колмогорова; $R$ — дійсна пряма, $Т$ — одиничне коло; $L_{p,s}^r (G)$ — множина функцій $x ∈ L_p(G)$ таких, що $x(r) ∈ L_s(G),\; q, p, s ∈ [1, ∞],\; k, r ∈ N,\; k < r$, $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} = 1 - k/r$$ якщо $K(R) = K(T)$, $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} < 1 - k/r$$ якщо $K(R) = K(T)$. Остання нерівність може бути як рівністю, так і строгою нерівністю. Як наслідок одержано нові точні нерівності типу Колмогорова на дійсній прямій. Institute of Mathematics, NAS of Ukraine 2003-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3934 Ukrains’kyi Matematychnyi Zhurnal; Vol. 55 No. 5 (2003); 579-589 Український математичний журнал; Том 55 № 5 (2003); 579-589 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3934/4581 https://umj.imath.kiev.ua/index.php/umj/article/view/3934/4582 Copyright (c) 2003 Babenko V. F.; Kofanov V. A.; Pichugov S. A.
spellingShingle Babenko, V. F.
Kofanov, V. A.
Pichugov, S. A.
Бабенко, В. Ф.
Кофанов, В. А.
Пичугов, С. А.
Бабенко, В. Ф.
Кофанов, В. А.
Пичугов, С. А.
Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle
title Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle
title_alt Сравнение точных констант в неравенствах для производных функций, заданных на вещественной оси и окружности
title_full Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle
title_fullStr Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle
title_full_unstemmed Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle
title_short Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle
title_sort comparison of exact constants in inequalities for derivatives of functions defined on the real axis and a circle
url https://umj.imath.kiev.ua/index.php/umj/article/view/3934
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