This paper presents an analysis of relativistic self-focusing of intense laser radiation in an axially inhomogeneous plasma. The nonlinearity in the dielectric constant arises on account of the relativistic variation of mass for an arbitrary magnitude of intensity. An appropriate expression for the nonlinear dielectric constant has been used in the analysis of laser-beam propagation in the paraxial approximation for a circularly polarized wave. The variations of the beam width parameter with the propagation distance, the self-trapping condition and the critical power have been evaluated. The saturating nature of the nonlinearity yields two values of the critical power for beam self-focusing, $P_{\rm cr1}$ and $P_{\rm cr2}$. It is seen that the laser beam width tends to attain a constant value depending upon the plasma inhomogenity and the initial laser intensity. Numerical estimates are made for typical values of the laser–plasma interaction applicable for underdense and overdense plasmas. The most important experiments described by this model are related to the fast ignitor concept.