A kinetic stability theorem is developed for relativistic non‐neutral electron flow in a planar high‐voltage diode with applied magnetic field. The effects of strong inhomogeneities and intense self‐electric and self‐magnetic fields are retained in the analysis in a fully self‐consistent manner. Use is made of global (spatially averaged) conservation constraints satisfied by the fully nonlinear Vlasov–Maxwell equations, assuming electromagnetic perturbations with extraordinary‐mode polarization, and space‐charge‐limited flow withE0x(x=0)=0 at the cathode. It is also assumed that they‐averaged,x‐directed net flux of particles,ymomentum, and energy, vanish identically at the cathode (x=0) and at the anode (x=d). It is shown that the class of self‐consistent Vlasov equilibriaf0b(H,Py) is stable for small‐amplitude perturbations, providedf0bis a monotonic decreasing function ofH−VbPy, i.e., provided ∂f0b/∂(H−VbPy)≤0. Here,His the energy andPyis the canonicalymomentum. The generality of thissufficientconditionforstabilityshould be emphasized. First, the derivation of the stability theorem has not been restricted to a specific choice off0b(H−VbPy). Moreover, the fully non‐neutral electron equilibria are generally characterized by strong spatial inhomogeneities and intense self‐electric and self‐magnetic fields. For the class of equilibria with ∂f0b/∂(H−VbPy)≤0, it is also shown that the density profilen0b(x)=∫ d3p f0bandx–xpressure profileP0b(x) =∫ d3p vxpxf0bdecrease monotonically from the cathode (x=0) to the anode (x=d) provided the applied magnetic field at the anode (Ba) is sufficiently strong that (Vb/c)Ba≥4&pgr;e ∫d0 dx’ n0b(x’).