On Chebyshev inequality for arithmetic functions
type: Item
instance of: scholarly article
title: On Chebyshev inequality for arithmetic functions
zbMATH Open document ID: 0684.10004
author: Jerzy Rutkowski
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 1989-01-01T00:00:00Z
review text: Using a multidimensional discrete version of the Chebyshev inequality the author obtains some inequalities concerning arithmetic functions illustrated with examples. One of his results is the following: An arithmetic function f is called d-non-decreasing (d-non-increasing) if for all positive integers m, n the inequality f(m)\(\leq f(mn)\) (f(m)\(\geq f(mn))\) holds. Now, let f and g be arithmetic functions, both d-non- decreasing or both d-non-increasing and let h be a multiplicative and non-negative arithmetic function. Then \[ \sum_{d | n}h(d)\sum_{d | n}h(d)f(d)g(d)\quad \geq \quad \sum_{d | n}h(d)f(d)\sum_{d | n}h(d)g(d) \] for every positive integer n. If one of the functions f and g is d-non-decreasing and the other d-non-increasing, then the sign \(\geq\) should be reversed.
reviewed by: László Tóth
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 4120234
zbMATH Keywords: multidimensional discrete version of the Chebyshev inequality
zbMATH Keywords: inequalities
zbMATH Keywords: arithmetic functions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4120234