A Unified Two-Parameter K–R Framework for Generalized Jensen, Hölder, Young, and Stability Inequalities

This paper introduces a unified two-parameter inequality framework governed by a pair of real constants (K,R), designed to generate generalized, refined, and stability-controlled variants of classical inequalities. The proposed K–R framework systematically extends fundamental results in mathematical analysis, including Jensen, Hölder, Young, integral, and discrete difference inequalities. By allowing independent dominance and perturbation control through the constants Kand R, the framework captures both contraction effects and stability deviations within a single analytic structure. Sharp K–R versions of Jensen and Hölder inequalities are established with complete proofs, together with K–R Young type product inequalities and K–R integral stability estimates of Poincaré–Sobolev type. Discrete K–R difference inequalities are further developed to analyze the stability and convergence of numerical schemes. Sharpness, equality conditions, and optimal parameter ranges are rigorously characterized. The proposed inequalities unify several scattered refinements in the literature and provide a flexible tuning mechanism for robustness analysis. Numerical simulations are presented to confirm the theoretical bounds and to illustrate the role of the R-parameter in stability enhancement. Applications to partial differential equation energy estimates, optimization under uncertainty, signal regularization, and iterative algorithms are discussed. The results demonstrate that the K–R framework offers a powerful and systematic approach for constructing new inequalities with built-in stability control, thereby enriching both continuous and discrete inequality theory.