We consider polynomial generalized Lienard systems that come from second order polynomial perturbations of a linear center. For these systems we found a generic (open and dense) subset in the space of perturbations of degree 2l, with l >= 1, such that each associated perturbed generalized Lienard system has at most 2l-1 medium limit cycles. Moreover, we show that this upper bound is reached. In addition, some concrete examples are given. (C) 2019 Elsevier Inc. All rights reserved.