Determinant Inequalities for Positive Definite Matrices Via Diananda’s Result for Arithmetic and Geometric Weighted Means

Silvestru Sever Dragomir

doi:10.2478/awutm-2023-0003

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Determinant Inequalities for Positive Definite Matrices Via Diananda’s Result for Arithmetic and Geometric Weighted Means

Annals of West University of Timisoara - Mathematics and Computer Science

Volume 59 (2023): Issue 1 (June 2023)

By: Silvestru Sever Dragomir

Open Access

|May 2023

Abstract

In this paper we prove among others that, if (A_j)_j=1,...,m are positive definite matrices of order n ≥ 2 and q_j ≥ 0, j = 1, ..., m with $\sum_{j = 1}^{m} q_{j} = 1$ $$\sum\nolimits_{j = 1}^m {{q_j} = 1} $$, then $\begin{array}{l} 0 & \leq \frac{1}{1 - \min_{i \in {1, \dots, m}} {q_{i}}} \\ \times & [{\sum^{}}_{i = 1}^{m} q_{i} (1 - q_{i}) {[\det (A_{i})]}^{- 1} - 2^{n + 1} \underset{1 \leq i < j \leq m}{\sum^{}} q_{i} q_{j} {[\det (A_{i} + A_{j})]}^{- 1}] \\ \leq & {\sum^{}}_{i = 1}^{m} q_{i} {[\det (A_{i})]}^{- 1} - {[\det ({\sum^{}}_{i = 1}^{m} q_{i} A_{i})]}^{- 1} \\ \leq & \frac{1}{\min_{i \in {1, \dots, m}} {q_{i}}} \\ \times & [{\sum^{}}_{i = 1}^{m} q_{i} (1 - q_{i}) {[\det (A_{i})]}^{- 1} - 2^{n + 1} \underset{1 \leq i < j \leq m}{\sum^{}} q_{i} q_{j} {[\det (A_{i} + A_{j})]}^{- 1}] . \end{array}$

References

E. F. Beckenbach, R. Bellman, Inequalities, Berlin-Heidelberg-New York, 1971.
Search in Google Scholar Back to article
P. H. Diananda, On some inequalities of H. Kober, Proc. Camb. Phil. Soc. 59 (1963), 341-346.
Search in Google Scholar Back to article
S. S. Dragomir, Determinant inequalities for positive definite matrices via Kober’s result for arithmetic and geometric weighted means, Preprint RGMIA Res. Rep. Coll. 24 (2021), Art. 148, 14 pp. https://rgmia.org/papers/v24/v24a148.pdf.
Search in Google Scholar Back to article
M. Ito, Estimations of the weighted power mean by the Heron mean and related inequalities for determinants and traces, Math. Inequal. Appl. 22, no. 3 (2019), 949-966.
Search in Google Scholar Back to article
Y. Li, L. Yongtao, Z. Huang Feng, W. Liu, Inequalities regarding partial trace and partial determinant, Math. Inequal. Appl. 23, no. 2 (2020), 477-485.
Search in Google Scholar Back to article
M. Lin, G. Sinnamon, Revisiting a sharpened version of Hadamard’s determinant inequality, Linear Algebra Appl. 606 (2020), 192-200.
Search in Google Scholar Back to article
J.-T. Liu, Q.-W. Wang, F.-F. Sun, Determinant inequalities for Hadamard product of positive definite matrices, Math. Inequal. Appl. 20, no. 2 (2017), 537-542.
Search in Google Scholar Back to article
W. Luo, Further extensions of Hartfiel’s determinant inequality to multiple matrices, Spec. Matrices 9 (2021), 78-82.
Search in Google Scholar Back to article
L. Mirsky, An inequality for positive definite matricies, Amer. Math. Monthly, 62 (1955), 428-430.
Search in Google Scholar Back to article
D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/awutm-2023-0003 | Journal eISSN: 1841-3307 | Journal ISSN: 1841-3293

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Language: English

Page range: 21 - 34

Published on: May 3, 2023

Published by: West University of Timisoara

In partnership with: Paradigm Publishing Services

Publication frequency: Volume open

Keywords:

Positive definite matrices,

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© 2023 Silvestru Sever Dragomir, published by West University of Timisoara
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 59 (2023): Issue 1 (June 2023)