In this article first, we give an integral identity and prove some Hermite-Hadamard type inequalities for the function f such that |f″|^qis convex or concave for q ≥ 1. Second, by using these results, we present applications to f-divergence measures. At the end, we obtain some bounds for special means of real numbers and new error estimates for the trapezoidal formula.

The main objective of the paper is to develop an innovative idea of bringing continuous and discrete inequalities into a unified form. The desired objective is thus obtained by embedding majorization theory with the existing notion of continuous inequalities. These notions are applied to the latest generalized form of the inequalities, popularly known as the Hermite–Hadamard–Jensen–Mercer inequalities. Moreover, the frequently-used Caputo fractional operators are employed, which are rightly considered critical, especially for applied problems. Both weighted and unweighted forms of the developed results are discussed. In addition to this, some bounds are also provided for the absolute difference between the left- and right-sides of the main results.

Automated decision-making systems are increasingly entrusted with life-defining decisions, yet prevailing fairness frameworks largely treat these systems as static, one-shot processes. This creates a critical gap: existing methods fail to capture how repeated decisions accumulate, allowing small disparities to evolve into lasting inequities. Fairness, however, is inherently temporal. A single loan denial can delay credit building for years, while repeated résumé rejections can systematically exclude qualified candidates. To address this limitation, a unified mathematical framework for temporal fairness is introduced, evaluating fairness not as a snapshot but as an evolving trajectory of opportunities and outcomes. The framework assesses whether individuals and groups progress at comparable rates or face persistent exclusion, revealing lock-out effects, widening opportunity gaps, and delays in access. Interpretable temporal metrics such as cumulative access, lock-out rate, and time-to-first-approval make long-term fairness measurable and auditable. Theoretical analysis and empirical validation demonstrate how feedback loops can either reinforce inequities or foster inclusion, and how short-term parity may conflict with long-term equity. By reframing fairness as a temporal property, this work provides both theoretical foundations and empirically grounded tools for developing decision systems that are transparent, adaptive, and sustainable in their equity commitments.

The Hermite–Hadamard–Mercer (HHM) inequality, existing in two well‐established forms, plays a fundamental role in mathematical analysis. This inequality is characterized by three distinct components—namely, the left, middle, and right terms. This study is concerned to obtain novel generalized and refined HHM fractional inequalities by employing for the first time the framework of Green functions. The interesting relationships among the three defining terms are investigated, and sharp bounds are derived for the differences between the left–middle and middle–right expressions. A key feature of this work is a curvature‐driven refinement of the HHM middle term, which systematically tightens the classical bounds and visibly narrows the gap between them. This refinement not only enhances the theoretical sharpness of the inequality but also provides a novel approach to how curvature and interval length influence the bounds. In addition, a thermal profile comparison between the classical and refined middle terms is included which show the improved sharpness of the obtained refinements. Also, error estimates are deduced for the difference of various means of real numbers by selecting appropriate functions and parameters within the obtained results.