Complete Bernstein Functions from Gamma Ratios with Square Root Arguments

We study Gamma ratios evaluated at a shifted square root and finite products of their positive powers. For each ratio, we prove the complete Bernstein property and compute the representing measure explicitly. The symmetry of the Gamma parameters gives a simple formula for the boundary phase, which determines when a product remains a complete Bernstein function. For orders of, at most, one half, the exact condition is that the weighted sum of the orders does not exceed one. For mixed orders, we give an exact boundary-phase criterion and an explicit sufficient condition, and we explain the role of complementary orders. We also determine the killing and drift coefficients, prove absolute continuity of the representing measures, and derive the corresponding Lévy densities. These formulas yield resolvent representations and subordinate semigroups for non-negative self-adjoint operators.