Oscillation for second-order differential equations with delay

We establishing monotonic properties of non-oscillatory solutions, and oscillation criteria for the second-order delay differential equation $$ y''(t)+p(t)y(\tau(t))=0. $$ The criteria obtained fulfil the gap in the oscillation theory and essentially improves the earlier ones. The progress is illustrated via Euler's differential equation. Moreover, we provide upper and lower bounds for the non-oscillatory solutions