Noninteger completely monotonic degrees of Stirling remainders and their derivatives

Abstract

For Stirling expansions truncated after at least two Bernoulli terms, we disprove the three real-degree conjectures recorded by Mahmoud and Qi for the signed remainders of the logarithm of the gamma function and their derivatives. We give a unified proof of the corresponding integer-degree formula, recovering known cases and resolving Zhu's conjectures for higher derivatives in this range. Every real completely monotonic degree lies strictly between the predicted integer and the next integer. We obtain a universal fractional lower bound through an explicit inverse function and a simpler rational bound valid for every derivative order. A local estimate for the negative part of the Laplace kernel's derivative with respect to its logarithmic variable makes the lower improvement proportional to the reciprocal of the combined truncation and derivative order. Together with a uniform upper bound, this determines the exact order of decay of the fractional excess for each fixed derivative order and for its supremum over all derivative orders. Sharper kernel estimates give stronger bounds for the first two degrees. The proof combines fractional positivity criteria, the classical Laplace representation, a sign characterization of the associated densities, and a convexity obstruction at the upper integer endpoint. All estimates used in the proofs are analytic. The exact real degrees and the optimal asymptotic constants remain open.