Pullback-Ratio Hermite–Hadamard–Fejér Certificates for Modified(h, m)–Convex Integral Averages under Nonlinear Kernel Generators

Abstract

This paper develops a pullback-ratio certificate framework for kernel-averaged Hermite Hadamard–Fejér type estimates under modified (h,m)–convexity of the second type. The central observation is that a nonlinear increasing generator does not preserve the ordinary affine parameter that appears in the generalized convexity inequality. Therefore the endpoint coefficients must be computed from the affine location induced by the generator pullback, rather than from the external kernel variable itself. This reparameterization yields valid scalar certificates for arbitrary admissible increasing generators, dimension-explicit vertex certificates on boxes, and radial certificates on simplices with the correct cone normalization. Matrix-path trace bounds are included only as a controlled application: the commuting case follows by simultaneous diagonalization, while the noncommuting statement is conditional on an explicit Loewner-order compatibility assumption. A new stochastic robustness section shows that the certificates remain stable under bounded random perturbations, zero-mean response noise, estimated generator ratios, and Monte Carlo validation error. Worked engineering-scale examples involving photovoltaic inverter losses, battery thermal loading, energy-portfolio allocation and covariance-barrier paths demonstrate how the corrected constants are computed in nontrivial applied settings. Unsupported assertions for arbitrary matrix means are deliberately avoided.