On functional inequalities and their applications in the oscillation theory

In the paper, we present several functional inequalities and offer their application for higher order advanced differential equations of the formx(n)(t)+q(t)x(σ(t))=0,x(n)(t)+q(t)x(σ(t))=0,to be oscillatory. The conditions obtained essentially improve, complements and extends many other known results and involves unimprovable constants.