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Opial’s Inequality

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Inequalities Involving Functions and Their Integrals and Derivatives

Part of the book series: Mathematics and Its Applications ((MAEE,volume 53))

Abstract

We devote this chapter to ideas surrounding the inequality of Z. Opial [1]. Theorem 1. If f ∈ C 1[0,a] with f(0) = f(a) = 0 and f(x) > 0 on (0,a), then

$$\int\limits_0^a {|f(x)f'(x)|dx \leqslant \frac{a}{4}} \int\limits_0^a {f'{{(x)}^2}dx} .$$
(1.1)

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Authors and Affiliations

  1. University of Belgrade, Yugoslavia

    D. S. Mitrinović

  2. University of Zagreb, Yugoslavia

    J. E. Pečarić

  3. Iowa State University, Ames, USA

    A. M. Fink

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  1. D. S. Mitrinović
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  2. J. E. Pečarić
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  3. A. M. Fink
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Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1991). Opial’s Inequality. In: Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications, vol 53. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3562-7_3

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  • DOI: https://doi.org/10.1007/978-94-011-3562-7_3

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