, where µ, τ are positive parameters and f is a strictly monotone, nonlinear C1-function satisfying f(0) = 0 and some convexity properties. It is well known that for prescribed oscillation frequencies (characterized by the
values of a discrete Lyapunov functional) there exists τ* > 0 such that for every τ > τ* there is a unique periodic solution. The period function is the minimal period of the unique periodic solution as a function
of τ > τ*. First we show that it is a monotone nondecreasing Lipschitz continuous function of τ with Lipschitz constant 2. As an application of our theorem we give a new proof of some recent results of Yi, Chen and Wu
 about uniqueness and existence of periodic solutions of a system of delay differential equations.