The problem of computing controllers to enlarge the domain of attraction (DA) of equilibrium points of polynomial systems is considered. In order to deal with such a problem, a technique for computing static nonlinear output feedback controllers, which maximize the largest estimate of the DA (LEDA) induced by a given polynomial Lyapunov function, is proposed. The main contribution of the note is to show that a lower bound of the maximum achievable LEDA and a corresponding controller can be obtained through linear matrix inequality optimizations. Moreover, a necessary condition for tightness of this lower bound is presented, which is also a sufficient condition to establish the tightness of the lower bound of the LEDA for a given controller.