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

Scalar inequalities in an order parameter often arise as the $2\times 2$ 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 applications include three named determinant lifts. First, we prove the positive Jackson $q$-gamma Hankel conjecture of Karp–Vishnyakova–Zhang: for $0<q<1$, the kernel $(x,y)\mapsto {\mathrm{\Gamma }}_q(x+y)$ is ${\mathrm{S}\mathrm{T}\mathrm{P}}_{\mathrm{\infty }}$. This is an atomic Mellin-moment instance of the general criterion; the reciprocal sign-regularity problem for $1\mathrm{/}{\mathrm{\Gamma }}_q$ is separate and is not addressed here. Second, we answer Yang’s continuous half-gamma Mills-ratio log-convexity question and strengthen it to strict total positivity, hence to all higher Hankel Turan determinants. Third, we treat Tricomi rays and the one-dimensional Coulomb regularization as all-minor Hankel determinant hierarchies. For the Coulomb regularization, the $2\times 2$ minor gives the scalar log-convexity question recorded by Baricz–Pogany, and the full theorem supplies the corresponding all-minor strengthening.