A Probabilistic Sign Rule for Quotients of Positive Series and Integral Transforms

This paper develops a probabilistic sign rule for quotients of functions represented by positive series or integrals. For a function in this class, normalising the summand function in the series case or the integrand function in the integral case induces a probability law under which parameter log-derivatives of the function are expressed as moments of kernels, the log-derivatives of the same summand or integrand function with respect to the same parameters. The resulting moment identities reduce quotient monotonicity, log-supermodularity, and log-convexity to sign criteria based on kernel monotonicity, stochastic ordering of the induced laws, and covariance or variance identities. The criteria are applied to generalised hypergeometric, Stieltjes-transform, and Prabhakar quotients, yielding new Tur\'an inequalities, two-sided Stieltjes bounds, and a local failure threshold for a monotonicity conjecture for the zero-balanced Gauss function.