Opial Inequalities

Ravi Agarwal, Donal O’Regan and Samir Saker
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Ravi Agarwal: Texas A&M University–Kingsville, Department of Mathematics
Donal O’Regan: National University of Ireland, School of Mathematics, Statistics, and Applied Mathematics
Samir Saker: Mansoura University, Department of Mathematics

Chapter Chapter 3 in Dynamic Inequalities On Time Scales, 2014, pp 93-174 from Springer

Abstract: Abstract In 1960 Opial proved that if x is absolutely continuous on [a, b] with x ( a ) = x ( b ) = 0 , $$x(a) = x(b) = 0,$$ then 3.0.1 ∫ a b x ( t ) x ′ ( t ) d t ≤ b − a 4 ∫ a b x ′ ( t ) 2 d t . $$\displaystyle{ \int _{a}^{b}\left \vert x(t)\right \vert \left \vert x^{{{\prime}} }(t)\right \vert dt \leq \frac{\left (b - a\right )} {4} \int _{a}^{b}\left \vert x^{{{\prime}} }(t)\right \vert ^{2}dt. }$$ We refer the reader to [9] for results on Opial type inequalities.

Keywords: Operator Inequality; Opial-type Inequalities; Lder Inequality; Classical Cauchy Schwarz Inequality; Higher Order Inequalities (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-3-319-11002-8_3

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