New monotonicity properties and oscillation of $n$-order functional differential equations with deviating argument

In this paper, we offer new technique for investigation of the even order linear differential equations of the form \begin{equation*}\label{E} y^{(n)}(t)=p(t)y(\tau(t)). \tag{$E$} \end{equation*} We establish new criteria for bounded and unbounded oscillation of \eqref{E} which improve a number of related ones in the literature. Our approach essentially involves establishing stronger monotonicities for the positive solutions of \eqref{E} than those presented in known works. We illustrate the improvement over known results by applying and comparing our technique with the other known methods on the particular examples