Refined Young Inequality and Its Application to Divergences
Shigeru Furuichi1, Nicuşor Minculete2
1Department of Information Science, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo 156-8550, Japan.
Entropy (Basel, Switzerland)
|April 30, 2021
Summary
This study provides new bounds for the difference between weighted arithmetic and geometric means, refining Young inequalities. These findings have applications in information theory, impacting various divergence measures.
Area of Science:
- Mathematical Analysis
- Information Theory
Background:
- The relationship between weighted arithmetic and geometric means is fundamental in inequalities.
- Young's inequality and its reverses are crucial in various mathematical fields.
- Divergence measures are essential for quantifying differences between probability distributions.
Purpose of the Study:
- To establish novel bounds for the difference between weighted arithmetic and geometric means.
- To derive refined versions of Young's inequality and its reverses.
- To explore the implications of these inequalities on several information-theoretic divergence measures.
Main Methods:
- Derivation of new inequalities bounding the difference between weighted means.
- Analysis of the properties of these newly established bounds.
- Application of the derived inequalities to analyze properties of specific divergence measures.
Main Results:
- New bounds on the difference between weighted arithmetic and geometric means were established.
- Refined Young inequalities and their reverses were obtained.
- Novel results concerning Tsallis, Rényi, Jeffreys-Tsallis, and Jensen-Shannon-Tsallis divergences were demonstrated.
Conclusions:
- The established bounds offer a deeper understanding of the relationship between weighted means.
- The refined inequalities have significant theoretical implications.
- The study provides new analytical tools for information-theoretic divergences.
Keywords:
Cartwright–Field inequalityHeinz meanJeffreys–Tsallis divergenceJensen–Shannon–Tsallis divergenceRényi divergenceTsallis divergenceYoung inequalityarithmetic meangeometric meanMore Related Videos
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