  In this paper, we introduce an inexact regularized proximal Newton method
(IRPNM) that does not require any line search. The method is designed to
minimize the sum of a twice continuously differentiable function $f$ and a
convex (possibly non-smooth and extended-valued) function $\varphi$. Instead of
controlling a step size by a line search procedure, we update the
regularization parameter in a suitable way, based on the success of the
previous iteration. The global convergence of the sequence of iterations and
its superlinear convergence rate under a local H\"olderian error bound
assumption are shown. Notably, these convergence results are obtained without
requiring a global Lipschitz property for $ \nabla f $, which, to the best of
the authors' knowledge, is a novel contribution for proximal Newton methods. To
highlight the efficiency of our approach, we provide numerical comparisons with
an IRPNM using a line search globalization and a modern FISTA-type method.
