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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Opial's inequality for $ f\sp {(n)}$


Author: A. M. Fink
Journal: Proc. Amer. Math. Soc. 115 (1992), 177-181
MSC: Primary 26D10
MathSciNet review: 1094500
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Abstract: We prove inequalities of the type

$\displaystyle {\int_0^h {\vert{f^{(i)}}(x){f^{(j)}}(x)\vert dx \leq C(n,i,j,p){... ...j + 1 - 2/p}}\left( {\int_0^h {\vert{f^{(n)}}(x){\vert^p}dx} } \right)} ^{2/p}}$

when $ f(0) = f'(0) = \cdots = {f^{(n - 1)}}(0) = 0$. We assume that $ {f^{(n - 1)}}$ is absolutely continuous and $ {f^{(n)}} \in {L_p}(0,h)$, with $ p \geq 1,n \geq 2$, and $ 0 \leq i \leq j \leq n - 1$.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1094500-7
PII: S 0002-9939(1992)1094500-7
Keywords: Opials's inequality
Article copyright: © Copyright 1992 American Mathematical Society