In this work, we present approximate analytical predictions for the contribution to the free energy of curvature of a thin (flexible) membrane rising from a polymer brush, which is grafted to both sides of the membrane. The influence of the approximations is revealed by a detailed comparison with numerically exact self-consistent field (SCF) calculations. We consider both the quenched case, i.e., when the grafting density is the same on both sides, and the annealed case, i.e., when the polymer chains can translocate upon bending from one side of the membrane to the other. It is found that the analytical predictions give the correct sign for the brush contribution to the free energy of curvature. Moreover, for spherically curved membranes, a reasonably accurate scaling with the grafting density or and the chain length N is obtained. However, in the case of a cylindrical curvature, the analytical models overestimate, the dependence on the polymer chain length. It is shown that the mean bending modulus is positive, which implies that the grafting of polymers onto membranes makes these stiffer. The Gaussian bending modulus is negative and scales with the chain length in the power three, whereas the mean bending modulus scales with the chain length with a power two. This is in contrast with the analytical predictions which point to the same power-law dependence of three. Our results imply that for sufficiently long polymers the flat conformation becomes unstable in favor of bending.