A polynomial-in-time growth bound is established for global Sobolev Hs(T) solutions to the derivative nonlinear Schrödinger equation on the circle with s> 1. These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution.
