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
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] .