A complementary inequality to the information monotonicity for Tsallis relative operator entropy

ABSTRACT

We establish a reverse inequality for Tsallis relative operator entropy involving a positive linear map. In addition, we present a converse of Ando's inequality, for each parameter. We give examples to compare our results with the known results by Furuta and Seo. In particular, we establish an extension and a reverse of the Löwner-Heinz inequality under certain conditions. Some interesting consequences in inner product spaces and norm inequalities are also presented.

KEYWORDS:

• Tsallis relative operator entropy
• Löwner-Heinz inequality
• positive operator
• positive linear map
• norm inequality

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary: 47A63
• Secondary: 47A30
• 47A64
• 46L05
• 47A12

Acknowledgments

The authors would like to thank the anonymous referees for valuable comments that have been implemented in the final version of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The author (S.F.) was partially supported by JSPS KAKENHI [grant number 16K05257].