Some quadratic integral inequalities of Opial type

We derive and investigate integral inequalities of Opial type: ${\displaystyle {\int }_{I}s|hḣ|dt\le {\int }_{I}rḣ²dt}$, where h ∈ H, I = (α,β) is any interval on the real line, H is a class of absolutely continuous functions h satisfying h(α) = 0 or h(β) = 0. Our method is a generalization of the method of [3]-[5]. Given the function r we determine the class of functions s for which quadratic integral inequalities of Opial type hold. Such classes have hitherto been described as the classes of solutions of a certain differential equation. In this paper a wider class of functions s is given which is the set of solutions of a certain differential inequality. This class is determined directly and some new inequalities are found.