On log-sum inequalities

ABSTRACT

In information theory, the well-known log-sum inequality is a fundamental tool which indicates the non-negativity for the relative entropy. In this article, we establish a set of inequalities which are similar to the log-sum inequality involving two functions defined on scalars. The parametric extended log-sum inequalities are shown. We extend these inequalities for the commutative matrices. In addition, utilizing the Löwner partial order relation and the Hansen–Pedersen theory for non-commutative positive semi-definite matrices we demonstrate a number of matrix-inequalities analogous to the log-sum inequality.

KEYWORDS:

• Log-sum inequality
• deformed logarithm
• convex function
• Löwner partial order
• Hansen–Pedersen theory

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 15A39
• 15A42
• 15A45

Acknowledgments

The authors would like to thank the referees for their careful and insightful comments to improve our manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Supriyo Dutta was a post-doctoral fellow at the Indian Institute of Technology Kharagpur during this work. Shigeru Furuichi was partially supported by JSPS KAKENHI [grant number 21K03341].