This paper develops a kernel-weighted framework of explicit vertex certificates for functions satisfying modified \((h,m)\)–convexity of the second type. Motivated by Hermite–Hadamard and Hermite–Hadamard–Fejer type integral bounds, the theory yields computable one-dimensional estimates with kernel expectations, dimension-explicit vertex certificates on rectangular domains through tensorized operators, and radial bounds on origin-anchored simplices via cone decompositions. Matrix counterparts are also obtained, including rigorous trace inequalities in the commuting regime and a conditional extension to noncommuting matrices under an operator-compatibility hypothesis. A central feature of the framework is that the associated kernel coefficients are explicit, one-dimensional, and reusable, so the resulting certificates can be verified directly in practice. Numerical investigations and illustrative computational examples show how the generalized parameters influence the admissible function class, the averaging geometry, and the conservatism of the resulting bounds. These examples also highlight the relevance of the framework to multivariate and matrix-valued applications that require conservative and numerically checkable certificates for integral averages.
This paper develops a kernel-weighted framework of explicit vertex certificates for functions satisfying modified \((h,m)\)–convexity of the second type. Motivated by Hermite–Hadamard and Hermite–Hadamard–Fejer type integral bounds, the theory yields computable one-dimensional estimates with kernel expectations, dimension-explicit vertex certificates on rectangular domains through tensorized operators, and radial bounds on origin-anchored simplices via cone decompositions. Matrix counterparts are also obtained, including rigorous trace inequalities in the commuting regime and a conditional extension to noncommuting matrices under an operator-compatibility hypothesis. A central feature of the framework is that the associated kernel coefficients are explicit, one-dimensional, and reusable, so the resulting certificates can be verified directly in practice. Numerical investigations and illustrative computational examples show how the generalized parameters influence the admissible function class, the averaging geometry, and the conservatism of the resulting bounds. These examples also highlight the relevance of the framework to multivariate and matrix-valued applications that require conservative and numerically checkable certificates for integral averages.
