The grand picture behind Jensen's inequality

TL;DR

This paper explores the underlying principles of Jensen's inequality by examining a broader class of inequalities involving functions and intervals, revealing a more general framework in mathematical analysis.

Contribution

It introduces a general condition involving functions and intervals that encompasses Jensen's inequality as a special case, offering a new perspective on convexity-related inequalities.

Findings

Provides a generalized inequality framework including Jensen's inequality

Shows the conditions under which certain function combinations lie within an interval

Offers insights into the structure of convexity and inequality relations

Abstract

Let $I$ and $J$ be two intervals, and let $f, g: I \rightarrow \mathbb{R}$. If for any points $a$ and $b$ in $I$ and any positive numbers $p$ and $q$ such that $p + q = 1$, we have \begin{align} \nonumber p f(a) + q f(b) + g(pa + qb) \in J, \end{align} then for any points $x_{1}, \ldots, x_{n}$ in $I$ and any positive numbers $\lambda_{1}, \ldots, \lambda_{n}$ such that $\sum_{i=1}^{n}\lambda_{i} = 1$, we have \begin{align} \nonumber \sum_{i=1}^{n}\lambda_{i} f(x_{i}) + g\left( \sum_{i=1}^{n}\lambda_{i}x_{i} \right) \in J. \end{align} If we take $g = -f$ and $J = [0, +\infty)$, then the Jensen's inequality. The conclusion is only a short glimpse of the grand picture behind Jensen's inequality shows in this paper.