Sharp inequalities and complete monotonicity for the Wallis ratio
type: Item
instance of: scholarly article
title: Sharp inequalities and complete monotonicity for the Wallis ratio
zbMATH Open document ID: 1209.26026
author: Cristinel Mortici
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2011-01-13T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1292334067
review text: If \(x>0\) and \(a\geq 0\) let \(f_a(x)=x! -(x-{1\over2})! -\log\sqrt{x+a}\). The author gives an elementary proof that if \(0\leq a\leq {1\over4}\) then \(f_a(x)\) is completely monotonic, while if \(a\geq {1\over2}\) then \(-f_a(x)\) is completely monotonic. Using this he then deduces that: \[ \sqrt{x+{1\over4}}<{x!\over(x-{1\over2})!}<{4\over\sqrt{5 \pi}}\sqrt{x+{1\over4}}, \quad\text{and}\quad 2\sqrt{{2\over3 \pi}}\sqrt{x+{1\over2}}<{x!\over(x-{1\over2})!}<\sqrt{x+{1\over2}} , \] where the constants being best possible. Other inequalities involving the digamma function are also obtained.
reviewed by: P. S. Bullen
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 26D07
zbMATH DE Number: 5835853
zbMATH Keywords: gamma function
zbMATH Keywords: digamma functions
zbMATH Keywords: polygamma functions
zbMATH Keywords: completely monotonic functions
zbMATH Keywords: Kazarinoff's inequality
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5835853