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2019 The Inequality of Milne and its Converse, III
Horst Alzer, Alexander Kovačec
Real Anal. Exchange 44(1): 89-100 (2019). DOI: 10.14321/realanalexch.44.1.0089

Abstract

The discrete version of Milne’s inequality and its converse states that \begin{equation*} (*)\quad \sum_{j=1}^n\frac{w_j}{1-p_j^2} \leq \sum_{j=1}^n\frac{w_j}{1-p_j} \sum_{j=1}^n\frac{w_j}{1+p_j} \leq \Bigl(\sum_{j=1}^n\frac{w_j}{1-p_j^2} \Bigr)^2 \end{equation*} is valid for all \(w_j>0\) \((j=1,...,n)\) with \(w_1+\dots+w_n=1\) and \(p_j\in (-1,1)\) \((j=1,...,n)\). We present new upper and lower bounds for the product \(\sum w/(1-p) \sum w/(1+p)\). In particular, we obtain an improvement of the right-hand side of \((*)\). Moreover, we prove a matrix analogue of our double-inequality.

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Horst Alzer. Alexander Kovačec. "The Inequality of Milne and its Converse, III." Real Anal. Exchange 44 (1) 89 - 100, 2019. https://doi.org/10.14321/realanalexch.44.1.0089

Information

Published: 2019
First available in Project Euclid: 27 June 2019

zbMATH: 07088965
MathSciNet: MR3951336
Digital Object Identifier: 10.14321/realanalexch.44.1.0089

Subjects:
Primary: 26D15
Secondary: 15A45

Rights: Copyright © 2019 Michigan State University Press

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Vol.44 • No. 1 • 2019
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