Using the first order averaging method for discontinuous differential system, the maximum number of limit cycles which bifurcate from the periodic annulus of the center for a class of generalized Lienard differential system is studied. By piecewise smooth polynomial perturbating, the linear estimation of the maximum number of limit cycles which bifurcate from the periodic annulus of this center is obtained. The result shows that there are more limit cycles which can bifurcate from the discontinuous Lienard differential system than the continuous one.