This article deals with asymptotic rejection of unmatched general periodic disturbances in nonminimum phase non-linear output feedback systems. The steady state responses are defined for unstable systems subject to general periodic disturbances, and the generalised gain and phase are defined for stable systems subject to general periodic disturbances. Based on the new definitions, new results are obtained for the equivalent input disturbance and disturbance estimation. An Lp-convergent estimate of the equivalent input disturbance is incorporated in the control design to ensure the asymptotic rejection of unmatched general periodic disturbances while maintaining the stability of the non-linear system.