A note on the period function for certain planar vector fields

Consider a smooth planar autonomous differential equation having a period annulus, P. We present a new criterion to ensure that the period function has at most one critical period on P. Our result has a compact form when the differential equation is written as ż = F(z, z̄). It is based on a suitable representation formula of the derivative of the period function which uses the infinitesimal generator associated to the continua of periodic orbits. We apply the criterion to several particular cases of the equation ż = f(z)g(z̄)/h(z, z̄), where f(z) and are holomorphic functions and h is a C2 smooth real valuated function. © 2010 Taylor & Francis.