On Opial’s inequality for $f^ {(n)}$
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- by A. M. Fink PDF
- Proc. Amer. Math. Soc. 115 (1992), 177-181 Request permission
Abstract:
We prove inequalities of the type \[ {\int _0^h {|{f^{(i)}}(x){f^{(j)}}(x)|dx \leq C(n,i,j,p){h^{2n - i - j + 1 - 2/p}}\left ( {\int _0^h {|{f^{(n)}}(x){|^p}dx} } \right )} ^{2/p}}\] 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
- Carl H. FitzGerald, Opial-type inequalities that involve higher order derivatives, General inequalities, 4 (Oberwolfach, 1983) Internat. Schriftenreihe Numer. Math., vol. 71, Birkhäuser, Basel, 1984, pp. 25–36. MR 821782
- D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686, DOI 10.1007/978-3-642-99970-3 D. S. Mitrinović, J. T. Pečarić, and A. M. Fink, Inequalities for functions involving their integrals and derivatives, Kluwer, Dordrecht, 1991.
- Z. Opial, Sur une inégalité, Ann. Polon. Math. 8 (1960), 29–32 (French). MR 112926, DOI 10.4064/ap-8-1-29-32
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 177-181
- MSC: Primary 26D10
- DOI: https://doi.org/10.1090/S0002-9939-1992-1094500-7
- MathSciNet review: 1094500