We study the stability of finite difference schemes for hyperbolic initial boundary value problems in one space dimension. Assuming stability for the dicretization of the hyperbolic operator as well as a geometric regularity condition, we show that the uniform Kreiss-Lopatinskii condition yields strong stability for the discretized initial boundary value problem. The present work extends results of Gustafsson, Kreiss, Sundstrom and a former work of ours to the widest possible class of finite difference schemes by dropping some technical assumptions. We give some new examples of numerical schemes for which our results apply.