On the period function for a family of complex differential equations

Abstract

We consider planar differential equations of the form z ˙ = f ( z ) g ( z ¯ ) being f ( z ) and g ( z ) holomorphic functions and prove that if g ( z ) is not constant then for any continuum of period orbits the period function has at most one isolated critical period, which is a minimum. Among other implications, the paper extends a well-known result for meromorphic equations, z ˙ = h ( z ) , that says that any continuum of periodic orbits has a constant period function.