  We prove enhanced binding and increase of binding energies in the non- and
semi-relativistic Pauli-Fierz models, for arbitrary values of the
fine-structure constant and the ultra-violet cut-off, and discuss the resulting
improvement of exponential localization of ground state eigenvectors. For the
semi-relativistic model we also discuss the increase of the renormalized
electron mass and determine the linear leading order term in the asymptotics of
the self-energy, as the ultra-violet cut-off goes to infinity.
