On some inequalities in the theory of uniform distribution. I
type: Item
instance of: scholarly article
title: On some inequalities in the theory of uniform distribution. I
zbMATH Open document ID: 0654.10044
author: Nedyalka A. Mitreva
author: Petko D. Proinov
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: The authors give a general theorem which states that for every one-sided Lipschitz function f on [0,1] with Lipschitz constant L and with \(f(0)=f(1)\), and for every nondecreasing, nonnegative function \(\phi\) on [0,\(\infty)\) one has that \(F(\| f\|)\) and \(2F([f])\) both are at most \(L \int^{1}_{0}\phi (| f(x)|)dx\) where \(F(x)=\int^{x}_{0}\phi (t) dt\) and \(\| f\|\) and [f] are the supremum norm and oscillation of f. This inequality can be applied to get results in the theory of uniform distribution as for example a generalization of LeVeque's inequality. [For Part II, see the following review Zbl 0654.10045)].
reviewed by: Gerhard Larcher
Mathematics Subject Classification ID: 11K06
zbMATH DE Number: 4067030
zbMATH Keywords: generalized discrepancy
zbMATH Keywords: one-sided Lipschitz function
zbMATH Keywords: inequality
zbMATH Keywords: uniform distribution
zbMATH Keywords: generalization of LeVeque's inequality
MaRDI profile type: Publication
full work available at URL: pjaa.64.80
OpenAlex ID: W2046076645
cites work: On distribution functions (mod 1): quantitative Fourier inversion
cites work: Q4767350
cites work: Q5511029
cites work: Q5682403
cites work: Quantitative versions of a result of Hecke in the theory of uniform distribution mod 1
cites work: Q4063530
cites work: Generalization of Two Results of the Theory of Uniform Distribution
cites work: Q3717150
cites work: On the Erdős-Turán inequality on uniform distribution. II
cites work: On Inequalities for Large Deviations for Certain Statistics
DOI: 10.3792/PJAA.64.80
description: scientific article; zbMATH DE number 4067030