Entropy: An inequality
type: Item
instance of: scholarly article
title: Entropy: An inequality
zbMATH Open document ID: 0685.28010
published in: Tokyo Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The authors provide a simple proof of an elementary inequality concerning entropy, which they have found useful in previous work on the Rudin- Shapiro sequence. Let \((p_ k| 0\leq k<\infty)\) be a probability vector satisfying for some \(\lambda >0,\quad \lambda p_ n\geq \sum^{\infty}_{k=n+1}p_ k\quad (n=0,1,2,...).\) The authors give a bound for \(\sum^{\infty}_{k=0}p_ k^{\alpha},\) with \(\alpha <1\) and utilize it to show \[ \sum^{\infty}_{k=0}p_ k \log(\frac{1}{p_ k})\leq \sum^{\infty}_{k=0}q_ k \log(\frac{1}{q_ k}), \] where \((q_ k)\) is a probability vector forming a geometric sequence for which \(\lambda q_ n=\sum^{\infty}_{k=n+1}q_ k\) \((n=0,1,2,...).\)
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 11B99
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 11K99
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 4122272
zbMATH Keywords: entropy of a probability distribution
zbMATH Keywords: Hölder inequality between
zbMATH Keywords: means
zbMATH Keywords: elementary inequality concerning entropy
zbMATH Keywords: Rudin-Shapiro sequence
author: Jean-Paul Allouche
author: Michel Mendès France
author: Gérald Tenenbaum
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270133978
OpenAlex ID: W2033736493
DOI: 10.3836/TJM/1270133978
description: scientific article; zbMATH DE number 4122272