  In this paper, we propose a generalized natural inflation (GNI) model to
study axion-like particle (ALP) inflation and dark matter (DM). GNI contains
two additional parameters $(n_1, n_2)$ in comparison with the natural
inflation, that make GNI more general. The $n_1$ build the connection between
GNI and other ALP inflation model, $n_2$ controls the inflaton mass. After
considering the cosmic microwave background and other cosmological observation
limits, the model can realize small-field inflation with a wide mass range, and
the ALP inflaton considering here can serve as the DM candidate for certain
parameter spaces.
