Functional Inequalities: New Perspectives and New Applications

TL;DR

This book presents a perspective that many functional inequalities arise from natural mathematical structures and physical phenomena, offering systematic methods for understanding, proving, and creating inequalities beyond traditional guesswork.

Contribution

It introduces a new viewpoint that connects functional inequalities to underlying structures, enabling systematic derivation and enhancement of inequalities.

Findings

Functional inequalities are linked to mathematical structures and physical phenomena.

Systematic approaches can be used to prove, understand, and improve inequalities.

New inequalities can be devised on demand using the proposed methods.

Abstract

This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. This project is rather an attempt at describing how most functional inequalities are not merely the byproduct of ingenious guess work by a few wizards among us, but are often manifestations of certain natural mathematical structures and physical phenomena. Our main goal here is to show how this point of view leads to "systematic" approaches for not just proving the most basic functional inequalities, but also for understanding and improving them, and for devising new ones - sometimes at will, and often on demand.