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We show that periods of solutions to Lipschitz functional differential equations cannot be too small. The problem on such periods is closely related to the unique solvability of the periodic value problem for linear functional differential equations. Sharp bounds for periods of non-constant solutions to functional differential equations with Lipschitz nonlinearities are obtained.doi:10.14232/ejqtde.2017.1.14 fatcat:fb572md7nfe6zltvwl24m5wobe