Refined Young Inequality and Its Application to Divergences

Shigeru Furuichi; Nicuşor Minculete

doi:10.3390/e23050514

Refined Young Inequality and Its Application to Divergences

Entropy (Basel). 2021 Apr 23;23(5):514. doi: 10.3390/e23050514.

Authors

Shigeru Furuichi¹, Nicuşor Minculete²

Affiliations

¹ Department of Information Science, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo 156-8550, Japan.
² Faculty of Mathematics and Computer Science, Transilvania University of Braşov, 500091 Brasov, Romania.

Abstract

We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also studied some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the Rényi divergence, the Jeffreys-Tsallis divergence and the Jensen-Shannon-Tsallis divergence.

Keywords: Cartwright–Field inequality; Heinz mean; Jeffreys–Tsallis divergence; Jensen–Shannon–Tsallis divergence; Rényi divergence; Tsallis divergence; Young inequality; arithmetic mean; geometric mean.