The Hermite–Hadamard inequality is a fundamental result in convex analysis, providing lower and upper bounds for the integral average of a convex function. This research aims to provide a unified treatment of Jensen-type and Hermite–Hadamard-type inequalities by utilizing various classes of Green functions. The study extends classical results to a broader context involving general measures and functional identities.
The core methodology involves the definition of several distinct Green functions. By employing these functions, integral identities are established for functions of the class C2. The study utilizes these identities to derive necessary and sufficient conditions for weighted Hermite-Hadamard inequalities to hold, specifically focusing on the behavior of the Green functions under integral operators.
The research establishes several major results. It proves the equivalence between the validity of generalized Jensen-type inequalities and specific integral conditions on Green functions. Furthermore, weighted versions of the Hermite–Hadamard inequality are derived, providing precise error bounds and refinements using Lp norms of the second derivative. New mean-value theorems of Lagrange and Cauchy types are also presented, characterizing the "intermediate point" in these inequalities.
This research shows how the application of Green functions provides a powerful and elegant framework for the systematic study of integral inequalities. The results unify several known generalizations of the Hermite–Hadamard inequality and offer new tools for error estimation in numerical analysis. This approach confirms that the classical Hermite–Hadamard estimates are specific cases of a much broader theory involving functional identities and second-order derivatives.
