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Wirtinger’s inequalities on time scales. (English) Zbl 1148.26020

Summary: This paper is devoted to the study of Wirtinger-type inequalities for the Lebesgue \(\Delta\)-integral on an arbitrary time scale \(T\). We prove a general inequality for a class of absolutely continuous functions on closed subintervals of an adequate subset of \(T\). By using this expression and by assuming that \(T\) is bounded, we deduce that a general inequality is valid for every absolutely continuous function on \(T\) such that its Delta-derivative belongs to \(L^2_{\Delta} ([a,b) \cap T)\) and at most it vanishes on the boundary of \(T\).

MSC:

26D15 Inequalities for sums, series and integrals
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