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2024 | OriginalPaper | Chapter

Matrix Inequalities for the Difference Between Arithmetic and Heinz Means

Authors : Fatimah Alsaafin, Aliaa Burqan

Published in: Mathematical Analysis and Numerical Methods

Publisher: Springer Nature Singapore

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Abstract

In this work, we provide new scalar and matrix inequalities that include the difference between the Heinz mean and the arithmetic mean. In specifically, we obtain the inequality.

\(\left( {\frac{{\text{p}}}{{\text{q}}}} \right)\left( {{\text{X}}\nabla {\text{Y}} - {\text{H}}_{{\text{q}}} \left( {{\text{X}},{\text{Y}}} \right)} \right) \le {\text{X}}\nabla {\text{Y}} - {\text{H}}_{{\text{p}}} \left( {{\text{X}},{\text{Y}}} \right) \le \left( {\frac{{1 - {\text{p}}}}{{1 - {\text{q}}}}} \right)\left( {{\text{X}}\nabla {\text{Y}} - {\text{H}}_{{\text{q}}} \left( {{\text{X}},{\text{Y}}} \right)} \right)\).

for any \({\text{n}} \times {\text{n }}\) complex positive definite matrices \({\text{X}}\) and \({\text{ Y}}\), where \({\text{ X}} < {\text{Y}}\) and \(0 < {\text{p}} \le {\text{q}} < 1\). Additionally, several determinant inequalities relevant to the Heinz and arithmetic means are given.

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Title: Matrix Inequalities for the Difference Between Arithmetic and Heinz Means
Authors: Fatimah Alsaafin
Aliaa Burqan
Publisher: Springer Nature Singapore
Book: Mathematical Analysis and Numerical Methods
Print ISBN: 978-981-9748-75-4

Electronic ISBN: 978-981-9748-76-1

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-981-97-4876-1_2

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