Revista Integración, temas de matemáticas.

Revista Integración, temas de matemáticas

Vol. 40 No. 2 (2022): Revista Integración, temas de matemáticas
Research and Innovation Articles

Determinant Inequalities for Positive Definite Matrices Via Additive and Multiplicative Young Inequalities

PDF
  • Silvestru Sever Dragomir
Silvestru Sever Dragomir
Victoria University
DOI: https://doi.org/10.18273/revint.v40n2-2022004

Published 2022-12-09

Keywords

  • Positive definite matrices,
  • Determinants,
  • Inequalities

How to Cite

Dragomir, S. S. (2022). Determinant Inequalities for Positive Definite Matrices Via Additive and Multiplicative Young Inequalities. Revista Integración, Temas De matemáticas, 40(2), 193–206. https://doi.org/10.18273/revint.v40n2-2022004

Copyright (c) 2022 Revista Integración, temas de matemáticas

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

In this paper we prove among others that, if the positive definite matrices A, B of order n satisfy the condition 0 < mIn ≤ B − A ≤ M In, for some constants 0 < m < M, where In is the identity matrix, then
0 ≤ (1 − t) [det (A)]−1 + t [det (A + mIn)]−1 − [det (A + mtIn)]−1
≤ (1 − t) [det (A)]−1 + t [det (B)]−1 − [det ((1 − t) A + tB)]−1
≤ (1 − t) [det (A)]−1 + t [det (A + M In)]−1 − [det (A + M tIn)]−1 ,
for all t ∈ [0, 1].

PDF

Downloads

Download data is not yet available.

References

  1. Beckenbach E. F. and Bellman R., Inequalities, Berlin-Heidelberg-New York, 1971.
  2. Bhatia R., ’Interpolating the arithmetic–geometric mean inequality and its operator version’, Lin. Alg. Appl. 413 (2006) 355–363.
  3. Kittaneh F. and Manasrah Y., "Improved Young and Heinz inequalities for matrix", J. Math. Anal. Appl. 361 (2010), 262-269.
  4. Kittaneh F. and Manasrah Y., "Reverse Young and Heinz inequalities for matrices", Lin. Multilin. Alg., 59 (2011), 1031-1037.
  5. Li Y., Yongtao L., Huang Feng Z.and Liu W., "Inequalities regarding partial trace and partial determinant". Math. Inequal. Appl. 23 (2020), no. 2, 477–485
  6. Lin M. and Sinnamon G., "Revisiting a sharpened version of Hadamard’s determinant inequality". Linear Algebra Appl. 606 (2020), 192–200.
  7. Liu J.-T., Wang Q.-W.and Sun F.-F., "Determinant inequalities for Hadamard product of positive definite matrices". Math. Inequal. Appl. 20 (2017), no. 2, 537–542.
  8. Luo W., "Further extensions of Hartfiel’s determinant inequality to multiple matrices". Spec. Matrices 9 (2021), 78–82.
  9. Ito M., "Estimations of the weighted power mean by the Heron mean and related inequalities for determinants and traces". Math. Inequal. Appl. 22 (2019), no. 3, 949–966.
  10. Mirsky L., "An inequality for positive definite matricies", Amer. Math. Monthly, 62 (1955), 428-430.
  11. Mitrinović D. S., Pečarić J. E. and Fink A.M., Classical and New Inequalities in Analysis, Kluwer Acedemi Publishers, 1993.