Positive definiteness of certain functions and comparison of power difference and Heinz means

Two families of operator means associated with power difference and Heinz means are considered. The condition (for parameters) under which Heinz means are majorized by power difference means in norms is well-understood, which is based on analysis of certain relevant ratios. We determine when the reversed inequalities hold true, which is the same as the positive definiteness question for reciprocal ratios. To achieve this goal, we study Fourier transforms via residue calculus. We also study positive definiteness for ( cosh ( a t ) cosh t + β ) - 1 with a ≥ 0 and β > - 1 by similar technique. Situation drastically changes at a = 1 / 2 .