Bounds of certain harmonic sums
type: Item
instance of: scholarly article
title: Bounds of certain harmonic sums
zbMATH Open document ID: 0873.40003
author: Juan-Miguel Gracia
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-10-26T00:00:00Z
review text: For fixed integers \(p,q,r\) and \(s\) with \(0<p<q\) and \(0\leq p+r< q+s\), define the constant \(\lambda= (ps-qr)/(q-p)\) and \[ \Phi(n) =(pn+ r+1)^{-1} +(pn+ r+2)^{-1}+ \cdots +(qn +s)^{-1} \] for each integer \(n\geq 1\). The main result shows that \(\Phi(n)\) is an increasing function of \(n\) if \(\lambda\leq {1\over 2}\), that \(\Phi(n)\) is a decreasing function if \(\lambda \geq 1\), and that \(\Phi(n)\) is increasing on \([0,n_0]\) and decreasing on \([n_0,\infty)\), for some \(n_0\geq 1\), if \(\lambda\in ({1\over 2},1)\). This result leads to upper and lower bounds for \(\Phi\), and extends the work of \textit{D. D. Adamović} and \textit{M. R. Tasković} [Publ. Fac. Electrotechn. Univ. Belgrade, Sér. Math. Phys. 247-273, 41-50 (1969; Zbl 0183.05001)], \textit{J. E. Pečarić} [Mat. Bilt. 5/6, 29-36 (1982; Zbl 0592.26018)] and \textit{S. Simić} [Mat. Vesn. 38, 331-336 (1986; Zbl 0648.40002)] in the cases where \(r=0=s\); \(r=0\), \(s=1\); and \(r=-1=s\).
Mathematics Subject Classification ID: 40A05
Mathematics Subject Classification ID: 40A25
zbMATH DE Number: 1003950
zbMATH Keywords: harmonic sums
zbMATH Keywords: series segments
zbMATH Keywords: monotone function
zbMATH Keywords: increasing function
zbMATH Keywords: decreasing function
zbMATH Keywords: bounds
reviewed by: R. James Tomkins
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5226
OpenAlex ID: W2086958735
DOI: 10.1006/JMAA.1997.5226
description: scientific article; zbMATH DE number 1003950