Finding and improving bounds of real functions by thermodynamic arguments

TL;DR

This paper explores how thermodynamic principles can be used to derive and improve bounds for real functions, such as the logarithm, by analyzing entropy production and irreversibility.

Contribution

It introduces a novel thermodynamic approach to establish and refine bounds for real functions, connecting physical entropy concepts with algebraic inequalities.

Findings

Derived bounds for the logarithmic function using thermodynamic methods

Established connections between entropy production and inequality bounds

Proposed new families of bounds for log(1+x) based on irreversibility levels

Abstract

The possibility of stating the second law of thermodynamics in terms of the increasing behaviour of a physical property establishes a connection between that branch of physics and the theory of algebraic inequalities. We use this connection to show how some well-known inequalities such as the standard bounds for the logarithmic function or generalizations of Bernoulli's inequality can be derived by thermodynamic methods. Additionally, we show that by comparing the global entropy production in processes implemented with decreasing levels of irreversibility but subject to the same change of state of one particular system, we can find progressively better bounds for the real function that represents the entropy variation of the system. As an application, some new families of bounds for the function $\log(1+x)$ are obtained by this method.