LAUSR

Abstract There is a close connection between stability and oscillation of delay differential equations. For the first-order equation x ′ ( t ) + c ( t ) x ( τ ( t ) ) = 0 , t ≥ 0 , where c is locally integrable of any sign, τ ( t ) ≤ t is Lebesgue measurable, lim t → ∞ ⁡ τ ( t ) = ∞ , we obtain sharp results, relating the speed of oscillation and stability. We thus unify the classical results of Myshkis and Lillo. We also generalise the 3/2-stability criterion to the case of measurable parameters, improving 1 + 1 / e to the sharp 3/2 constant.