Real Analysis Exchange

The Inequality of Milne and its Converse, III

Horst Alzer and Alexander Kovačec

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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.

Article information

Real Anal. Exchange, Volume 44, Number 1 (2019), 89-100.

First available in Project Euclid: 27 June 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 15A45: Miscellaneous inequalities involving matrices

Milne's inequality matrix inequalities


Alzer, Horst; Kovačec, Alexander. The Inequality of Milne and its Converse, III. Real Anal. Exchange 44 (2019), no. 1, 89--100. doi:10.14321/realanalexch.44.1.0089.

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