  We study growth of higher Sobolev norms of solutions to the one-dimensional
periodic nonlinear Schrodinger equation (NLS). By a combination of the normal
form reduction and the upside-down I-method, we establish \|u(t)\|_{H^s}
\lesssim (1+|t|)^{\alpha (s-1)+} with \alpha = 1 for a general power
nonlinearity. In the quintic case, we obtain the above estimate with \alpha =
1/2 via the space-time estimate due to Bourgain [4], [5]. In the cubic case, we
concretely compute the terms arising in the first few steps of the normal form
reduction and prove the above estimate with \alpha = 4/9. These results improve
the previously known results (except for the quintic case.) In Appendix, we
also show how Bourgain's idea in [4] on the normal form reduction for the
quintic nonlinearity can be applied to other powers.
