Order-Moment Transport and Hankel Determinants in Special-Function Inequalities

Scalar inequalities in an order parameter often arise as the $2\times2$ shadow of a stronger Hankel determinant statement. We record a moment-representation criterion: positive exponential and Mellin order representations, together with gamma-normalized completely monotone averages, generate totally nonnegative Hankel kernels, with strictness controlled by the support of the representing measure. The criterion packages the classical total-positivity mechanism as a recognition calculus for special-function inequalities, turning the order parameter into a moment exponent after the correct normalization.
The main application answers Yang's continuous half-gamma Mills-ratio log-convexity question and strengthens it to strict total positivity, hence to all higher Hankel Turán determinants. A second application treats Tricomi rays and the one-dimensional Coulomb regularization as all-minor Hankel determinant hierarchies. For the Coulomb regularization, the $2\times2$ minor gives the scalar log-convexity question recorded by Baricz--Pogány, and the full theorem supplies the corresponding all-minor strengthening.