Finding and improving bounds of real functions by thermodynamic arguments

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.