<p>This paper focuses on refining the Jensen–Mercer inequality and extending its applications to various important inequalities, including Hölder’s, Ky Fan, and AM-GM inequalities. The refinements made to these classical inequalities lead to more precise bounds, offering enhanced capabilities for error estimation, optimization, and performance guarantees. Additionally, the paper explores the application of these refined inequalities to divergence measures such as the Csiszár divergence, Kullback–Leibler divergence, Shannon entropy, and others. These measures play a crucial role in information theory, machine learning, and statistical analysis.</p>