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An Inequality Ascribed to Wirtinger and Related Results

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Inequalities Involving Functions and Their Integrals and Derivatives

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

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Abstract

In this Chapter we shall consider inequalities of the form

$$\parallel f{\parallel _p} \leqslant C(p,q,n){(b - a)^{n - \frac{1}{p} - \frac{1}{q}}}\parallel {f^{(n)}}{\parallel _q}$$
(1.1)

on an interval [a, b] under the assumptions that f (n-1) is absolutely continuous and f (n) is in L q. The exponent of (b — a) is selected to make the inequality invariant under translation of the interval and invariant under scale change, as it must be. We attempt as much as possible to get the best possible constant and reserve the notation C(p, q, n) for the best possible constant. Now, such an inequality as (1.1) does not exist for all functions f since the kernel of the map from f to f (n) is n-dimensional. Hence in order to get such an inequality we must impose at least n independent conditions on f.

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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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Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1991). An Inequality Ascribed to Wirtinger and Related Results. 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_2

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