Method of Separating Tangents

TL;DR

This paper extends Jensen's inequality to certain nonconvex functions by analyzing their local geometric properties, enabling broader applications in various scientific fields.

Contribution

It introduces a novel method for applying Jensen's inequality to nonconvex functions based on their local geometric positioning relative to a base curve.

Findings

Applicable to about 75% of studied inequalities related to Jensen's inequality.

Allows proving complex inequalities more easily.

Broadens the scope of Jensen's inequality in physics, economy, and information theory.

Abstract

The well known Jensen inequality, holds true for every convex functions. However, we found that it is possible to apply it to some problems related to nonconvex functions for which Jensen's inequality holds true locally. Having considered a set of such functions, we noted some general patterns. We show that the key point, which provides Jensen's inequality holds true locally, is that the plot of function should be situated at only one side from the local base curve defined compatible with conditional variables. Moreover, we have achieved even more general result. It turned out that the graph of the function can be located on either sides of the local base curve, with the conditions. This result allows one to prove easily difficult types of inequalities, and on the other hand to broaden applications in physics, economy, and information theory. On the basis of the conducted analysis of different sources it is possible to claim, that our method is applicable to about three fourths of studied inequalities related to Jensen's inequality.