Nihonkai Mathematical Journal

The Quadratic Quantum $f$-Divergence of Convex Functions and Matrices

Silvestru Sever Dragomir

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In this paper we introduce the concept of quadratic quantum $f$-divergence measure for a continuos function $f$ defined on the positive semi-axis of real numbers, the invertible matrix $T$ and matrix $V$ by $$\mathcal{S}_{f}\left( V,T\right) :=\mathrm{tr}\left[ \left\vert T^{\ast }\right\vert ^{2}f\left( \left\vert VT^{-1}\right\vert ^{2}\right) \right].$$ Some fundamental inequalities for this quantum $f$-divergence in the case of convex functions are established. Applications for particular quantum divergence measures of interest are also provided.


The author would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of the paper.

Article information

Nihonkai Math. J., Volume 29, Number 1 (2018), 1-19.

Received: 13 October 2016
Revised: 15 February 2018
First available in Project Euclid: 6 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A63: Operator inequalities 47A30: Norms (inequalities, more than one norm, etc.)
Secondary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 26D15: Inequalities for sums, series and integrals 26D10: Inequalities involving derivatives and differential and integral operators

operator perspective convex functions operator inequalities arithmetic mean-geometric mean operator inequality relative operator entropy


Dragomir, Silvestru Sever. The Quadratic Quantum $f$-Divergence of Convex Functions and Matrices. Nihonkai Math. J. 29 (2018), no. 1, 1--19.

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