We study models of quartic inflation where the inflaton field phgr is coupled non-minimally to gravity, ξ phgr2 R, and perform a study of quantum corrections in curved space-time at one-loop level. We specifically focus on comparing results between the metric and Palatini theories of gravity. Transformation from the Jordan to the Einstein frame gives different results for the two formulations and by using an effective field theory expansion we derive the appropriate β-functions and the renormalisation group improved effective potentials in curved space for both cases in the Einstein frame. In particular, we show that in both formalisms the Einstein frame depends on the order of perturbation theory but that the flatness of the potential is unaltered by quantum corrections.