DETERMINANT INEQUALITIES FOR POSITIVE DEFINITE MATRICES VIA BHATIA AND KITTANEH-MANASRAH RESULTS

SILVESTRU SEVER DRAGOMIR
Mathematics, College of Engineering & Science Victoria University, PO Box 14428 Melbourne City, MC 8001, Australia. e-mail: sever.dragomir@vu.edu.au
and
DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa URL: http://rgmia.org/dragomir

 Issue: SSRSMI, Number 1, Volume XXXII Section: Volume 32, Number 1 Abstract: In this paper we prove among others that, if A and B are positive definite matrices, then [the following inequality holds]: 0& \leq \int_{0}^{1}\left[ \det \left( \left( 1-t\right) A+tB\right) \right] ^{-1}dt-\left[ \det \left( \frac{A+B}{2}\right) \right] ^{-1} \\ & \leq \frac{1}{3}\left[ \frac{1}{2}\left( \left[ \det \left( A\right) % \right] ^{-1}+\left[ \det \left( B\right) \right] ^{-1}\right) -\left[ \det \left( \frac{A+B}{2}\right) \right] ^{-1}\right] \\ & \leq \frac{1}{2}\left( \left[ \det \left( A\right) \right] ^{-1}+\left[ \det \left( B\right) \right] ^{-1}\right) -\int_{0}^{1}\left[ \det \left( \left( 1-t\right) A+tB\right) \right] ^{-1}dt \\ & \leq \frac{4}{3}\left[ \frac{1}{2}\left( \left[ \det \left( A\right) % \right] ^{-1}+\left[ \det \left( B\right) \right] ^{-1}\right) -\left[ \det \left( \frac{A+B}{2}\right) \right] ^{-1}\right] . Keywords: Code [ID]: SSRSMI202201V32S01A0003 [0005518] Note: DOI: Full paper: Download pdf

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