Refined Young Inequality and Its Application to Divergences

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

• Young inequality
• arithmetic mean
• geometric mean
• Heinz mean
• Cartwright–Field inequality
• Tsallis divergence