Monotonous period function for equivariant differential equations with homogeneous nonlinearities

We prove that the period function of the center at the origin of the ${\mathbb{Z}}_k$-equivariant differential equation $\dot{z}=iz+a(z\bar{z}{)}^n{z}^{k+1},a\mathrm{\ne }0,$ is monotonous decreasing for all $n$ and $k$ positive integers, solving a conjecture about them. We show this result as corollary of proving that the period function of the center at the origin of a sub-family of the reversible quadratic centers is monotonous decreasing as well.