  In this work we shall study a new class of attractor models which we shall
call generalized $R^p$-attractor models. This class of models is based on a
generalization of the Einstein frame potential of $R^p$ $f(R)$ gravity models
in the Jordan frame. We present the attractor properties of the corresponding
non-minimally coupled Jordan frame theory, and we calculate the observational
indices of inflation in the Einstein frame. As we show, there is a large class
of non-minimally coupled scalar theories, with an arbitrary non-minimal
coupling which satisfies certain conditions, that yield the same Einstein frame
potential, this is why these models are characterized attractors. As we
demonstrate, the generalized $R^p$-attractor models are viable and well fitted
within the Planck constraints. This includes the subclass of the generalized
$R^p$-attractor models, namely the Einstein frame potential of $R^p$ inflation
in the Jordan frame, a feature also known in the literature. We also highlight
an important issue related to the $R^p$ inflation in the Jordan frame, which is
known to be non-viable. By conformal invariance, the $R^p$ inflation model
should also be viable in the Jordan frame, which is not. We pinpoint the source
of the problem using two different approaches in the $f(R)$ gravity Jordan
frame, and as we demonstrate, the problem arises in the literature due to some
standard simplifications made for the sake of analyticity. We demonstrate the
correct way to analyze $R^p$ inflation in the Jordan frame, using solely the
slow-roll conditions.
