Geodesic
Some quadratic integral inequalities of Opial type
Annales Polonici Mathematici, Tome 63 (1996) no. 2, pp. 103-113

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We derive and investigate integral inequalities of Opial type: $∫_I s|hḣ|dt ≤ ∫_I rḣ² dt$, 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.
DOI : 10.4064/ap-63-2-103-113
Keywords: absolutely continuous function, integral inequality 
Małgorzata Kuchta. Some quadratic integral inequalities of Opial type. Annales Polonici Mathematici, Tome 63 (1996) no. 2, pp. 103-113. doi: 10.4064/ap-63-2-103-113
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