Functional-Analytic Inequalities and Matrix-Valued Energy Certificates for Convex Optimization

We establish explicitly conditioned multivariate Hermite–Hadamard–Fejér (HH–F) upper certificates for modified (h,m)–convexity on origin-anchored boxes and simplices, together with operator and matrix counterparts that yield trace- and eigenvalue-level certificates. To keep the theory mathematically consistent, the scalar and box analyses are organized into three rigorous regimes: the identity-generator case, the affine-generator case, and a genuinely nonlinear weighted case treated through w-adapted convexity along the pullback path induced by the kernel generator. The simplex analysis is presented in two layers, namely a standard radial theorem and a weighted radial extension. Within this structure, we introduce tensorized kernel-weighted integral operators generated by coordinate-wise maps wi and Fejér-symmetric kernels, recover classical rectangle bounds as limiting cases, and obtain dimension-explicit vertex certificates on boxes together with mixed facet and vertex certificates on simplices. In the operator setting, the commuting regime is fully rigorous by simultaneous diagonalization and functional calculus, whereas the noncommuting regime is stated only under explicit operator-compatibility hypotheses. Verified nonlinear pullback and simplex examples, an explicit certificate-screening step, and dimension-dependent cost calculations demonstrate the reach and limitations of the bounds. Engineering illustrations are analytic proof-of-concept calculations; previously reported empirical data-set tables lacking the requisite scripts and split indices have been withdrawn from the main text.