The paper establishes the nature of the time function (which turns out to be a distribution) that corresponds to a Laplace transform that is bounded and holomorphic in a half plane. This relationship is used to generalise the contraction-mapping stability criterion for constant linear multivariable systems. It is shown that the generalisation, which has the same frequency response interpretation as Rosenbrock's diagonal-dominance criterion, includes the cases considered earlier by Desoer and Wu (‘Stability of multiple-loop feedback linear time invariant systems’, J. Maths. Anal. & Appl., 1968, 23, pp. 121–129) and Rosenbrock [‘Design of multivariable control systems using the inverse Nyquist array’, Proc. IEE, 1969, 116, (11), pp. 1929–1936] as particular cases.