标题： 从热力学得到启发的广义詹森不等式 显示英文标题

标题： Generalized Jensen's inequality motivated from thermodynamics

摘要： 在本文中，我们通过提供一个不等式来推广P.T.Landsberg\cite{web1,web2}和S.S.Sidhu\cite{web3}的工作，该不等式的主要动机来自热力学定律，以定理的形式呈现，该定理在生成不同的不等式方面非常有用，例如加权AM-GM-HM不等式、p次幂不等式、Jensen不等式和其他许多不等式。在本文中，我们不仅给出了该不等式的热力学背景，还以一种直接严格的证明形式提供了必要的数学论证，使用了基本实分析，而这一点在Landsberg和Sidhu的工作中并未出现。 事实上，定理的第一个陈述从数学上证明了当n个不同温度的物体接触时所达到的平衡温度的唯一性。 定理的第二个陈述从数学上证明了这样一个过程是自发的，即熵有利的：当n个不同温度的物体接触时，它们会达到一个共同的温度。 因此，本文鼓励学生通过观察自然界已有的现象来提出不同的数学结果，并帮助他们通过将相关的物理现象与这些不等式联系起来，来欣赏他们在中学和大学阶段学到的传统不等式。 显示英文摘要

摘要： 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.