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In this work we shall study a new class of attractor models which we shallcall generalized $R^p$-attractor models. This class of models is based on ageneralization of the Einstein frame potential of $R^p$ $f(R)$ gravity modelsin the Jordan frame. We present the attractor properties of the correspondingnon-minimally coupled Jordan frame theory, and we calculate the observationalindices of inflation in the Einstein frame. As we show, there is a large classof non-minimally coupled scalar theories, with an arbitrary non-minimalcoupling which satisfies certain conditions, that yield the same Einstein framepotential, this is why these models are characterized attractors. As wedemonstrate, the generalized $R^p$-attractor models are viable and well fittedwithin the Planck constraints. This includes the subclass of the generalized$R^p$-attractor models, namely the Einstein frame potential of $R^p$ inflationin the Jordan frame, a feature also known in the literature. We also highlightan important issue related to the $R^p$ inflation in the Jordan frame, which isknown to be non-viable. By conformal invariance, the $R^p$ inflation modelshould also be viable in the Jordan frame, which is not. We pinpoint the sourceof the problem using two different approaches in the $f(R)$ gravity Jordanframe, and as we demonstrate, the problem arises in the literature due to somestandard simplifications made for the sake of analyticity. We demonstrate thecorrect way to analyze $R^p$ inflation in the Jordan frame, using solely theslow-roll conditions.

Generalized $R^p$-attractor Cosmology in the Jordan and Einstein Frames: New Type of Attractors and Revisiting Standard Jordan Frame $R^p$ Inflation