We consider the classical problem of absolute stability, i.e. stability of a linear time-invariant system in feedback with a memoryless time-varying sector bound non-linearity. We derive necessary and sufficient conditions for the existence of periodic motion. For the special case of symmetric sector bounds, this condition yields the minimum sector bounds for which the system can sustain periodic motion. By guaranteeing the absence of periodic motion we derive non-conservative sufficient conditions for absolute stability for arbitrary sector bounds. We demonstrate by numerical examples the application of our results.