the monotonicity and convexity of the period function for a class of symmetric newtonian systems of degree 8

in this paper, we study the monotonicity and convexity of the period function associated with centers of a speci c class of symmetric newtonian systems of degree 8. in this regard, we prove that if the period annulus surrounds only one elementary center, then the corresponding period function is monotone; but, for the other cases, the period function has exactly one critical point. we also prove that in all cases, the period function is convex.