  We present an extended study of inflationary models that inflaton field is
non-minimally coupled with gravity. We study parameters space of the models up
to the second (and in some cases third) order of the slow-roll parameters for
usual large field potentials in Jordan and Einstein frames that are connected
each other by conformal transformation. We calculate inflationary parameters
and the results are compared in both frames and also with observations. By
using the recent observational datasets, we present a discussion in order to
clarify the physical frame between Jordan and Einstein frames. Also, some
suggestions are expressed in order to navigate us for the future works.
