Version 1
: Received: 13 December 2019 / Approved: 15 December 2019 / Online: 15 December 2019 (14:08:32 CET)
Version 2
: Received: 11 January 2020 / Approved: 12 January 2020 / Online: 12 January 2020 (17:55:39 CET)
How to cite:
Kian, M.; Alomari, M.W. Klein's Trace Inequality and Superquadratic Trace Functions. Preprints2019, 2019120192. https://doi.org/10.20944/preprints201912.0192.v2
Kian, M.; Alomari, M.W. Klein's Trace Inequality and Superquadratic Trace Functions. Preprints 2019, 2019120192. https://doi.org/10.20944/preprints201912.0192.v2
Kian, M.; Alomari, M.W. Klein's Trace Inequality and Superquadratic Trace Functions. Preprints2019, 2019120192. https://doi.org/10.20944/preprints201912.0192.v2
APA Style
Kian, M., & Alomari, M.W. (2020). Klein's Trace Inequality and Superquadratic Trace Functions. Preprints. https://doi.org/10.20944/preprints201912.0192.v2
Chicago/Turabian Style
Kian, M. and Mohammad W. Alomari. 2020 "Klein's Trace Inequality and Superquadratic Trace Functions" Preprints. https://doi.org/10.20944/preprints201912.0192.v2
Abstract
We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{{A + B}}{2}} \right) +\tr f\left(\left| {\frac{{A - B}}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*} holds for all positive matrices $A,B$. In addition, we present a Klein's inequality for superquadratic functions as $$ \mathrm{Tr}[f(A)-f(B)-(A-B)f'(B)]\geq \mathrm{Tr}[f(|A-B|)] $$ for all positive matrices $A,B$. It gives in particular improvement of Klein's inequality for non-negative convex function. As a consequence, some variants of the Jensen trace inequality for superquadratic functions have been presented.
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Commenter: Mohammad Alomari
Commenter's Conflict of Interests: Author