About the sharpness of the Jensen inequality

Area of Science:

• Mathematical Analysis
• Information Theory

Background:

• Jensen's inequality is a fundamental concept in convex analysis.
• Recent advancements have focused on the sharpness of this inequality.

Purpose of the Study:

• To improve existing results concerning the sharpness of Jensen's inequality.
• To extend the analysis to real Stieltjes measures that are not necessarily positive.

Main Methods:

• Utilizing novel Green functions for analysis.
• Investigating the properties of non-positive real Stieltjes measures.

Main Results:

• An improved bound on the sharpness of Jensen's inequality.
• Demonstration of the applicability of the refined inequality.

Conclusions:

• The presented method offers a more precise understanding of Jensen's inequality's sharpness.
• The findings have implications for information theory and statistical modeling.