Generalized Jensen's inequality motivated from thermodynamics

TL;DR

The paper introduces a thermodynamics-inspired generalization of Jensen-type and related inequalities. By defining $F(x)=\sum_{i=1}^n \int_{x_i}^{x} f_i(t)\,dt$ and a unique equilibrium temperature $x_0$ with $F(x_0)=0$, it proves an inequality framework: $\sum_{i=1}^n \int_{x_i}^{x_0} f_i(x) g(x_0) \le \sum_{i=1}^n \int_{x_i}^{x_0} f_i(x) g(x)$ for decreasing $g$, and the reverse for increasing $g$, relying only on integrability and monotonicity. This leads to a rigorous derivation of Jensen’s inequality for differentiable $F$, and yields traditional results such as the weighted AM-GM-HM and $p$-th power inequalities as special cases, plus additional non-Jensen-type inequalities. The work offers a physics-grounded, elementary pathway to a broad family of classical inequalities, with pedagogical and theoretical implications for deriving mathematical results from natural phenomena.

Abstract

In this paper, we generalize the work of P.T.Landsberg\cite{web1,web2} and S.S.Sidhu\cite{web3} by providing an inequality that has its main motivation from the laws of thermodynamics, in the form of a theorem which is quite useful in generating different inequalities such as the weighted AM-GM-HM inequality, the p-th power inequality , Jensen's inequality and many other inequalities.In this paper, we have not only given the thermodynamic motivation behind the inequality but we have given the required mathematical justification in the form of a straightforward rigorous proof using basic real analysis , which was not present in the works of Landsberg and Sidhu. In fact, the first statement of the theorem mathematically proves the uniqueness of the equilibrium temperature that is attained when n different bodies at different temperatures are brought in contact. The second statement of the theorem gives a mathematical proof of the fact that the process in which n bodies at different temperatures when brought in contact equilibriate to a common temperature is spontaneous,i.e., entropically favourable. Thus, this article motivates the students to come up with different mathematical results by observing the phenomena already existing in nature and also helps them to appreciate the conventional inequalities taught to them at the secondary school and undergraduate level by associating relevant physical phenomena with those inequalities.