Period function of planar turning points

This paper is devoted to the study of period function planar generic and non-generic turning points. In case (resp. non-generic) a non-degenerate degenerate) center disappears in limit e → 0, where ≥ 0 singular perturbation parameter. We show that, for each > ∼ monotonously increasing has exactly one minimum). The result valid an e-uniform neighborhood also solve part conjecture about uniform upper bound number critical periods inside classical Lienard systems fixed degree, formulated by De Maesschalck Dumortier 2007. use theory family blow-up.