Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

# Klein's Trace Inequality and Superquadratic Trace Functions

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. Preprints 2019, 2019120192 (doi: 10.20944/preprints201912.0192.v2). Kian, M.; Alomari, M.W. Klein's Trace Inequality and Superquadratic Trace Functions. Preprints 2019, 2019120192 (doi: 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.

## Keywords

Klein's inequality; Jensen's trace inequality; trace inequality

## Subject

MATHEMATICS & COMPUTER SCIENCE, Analysis

Comment 1
Commenter's Conflict of Interests: Author
Comment: We mainly revised our manuscript and correct some results and add new ones. A new author is added.
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