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An elementary proof of Gorny's inequality

Published online by Cambridge University Press:  14 November 2011

C. Fabry
C. Fabry
Affiliation:
Institut Mathematique, Université Catholique de Louvain, chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium
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Synopsis

Gorny's inequality provides upper bounds for the sup-norms ∥f(k) of a function f over an interval [a, b] for k = 1, …, n − 1, assuming the sup-norms of f and f(n) to be given. We present a simple proof of that inequality and obtain sharper estimates of the constants contained in that inequality, compared with the original verison of Gorny.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 105 , Issue 1 , 1987 , pp. 345 - 349
Copyright
Copyright © Royal Society of Edinburgh 1987

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References

1Cavaretta, A.S. Jr., An elementary proof of Kolmogorov's theorem. Amer. Math. Monthly 81 (1974), 480–486.CrossRefGoogle Scholar
2Duffin, R. J. and Schaeffer, A. C.. A refinement of an inequality of the brothers Markoff. Trans. Amer. Math. Soc. 50 (1941), 517–528.CrossRefGoogle Scholar
3Eloe, P. W. and Henderson, J.. Nonlinear boundary value problems and a priori bounds on solutions. SIAM J. Math. Anal. 15 (1984), 642–647.Google Scholar
4Gorny, A.. Contribution à l'étude des fonctions dérivables d'une variable réelle. Ada Math. 71 (1939), 317–358.Google Scholar
5Jackson, L. K.. A Nagumo condition for ordinary differential equations. Proc. Amer. Math. Soc. 57 (1976), 93–96.CrossRefGoogle Scholar
6Karlin, S.. Oscillatory perfect splines and related extremal problems. In Studies in Spline Functions and Approximation Theory, eds. Karlin, S., Micchelli, C. A., Pinkus, A. and Schoenberg, I. J. (New York: Academic Press, 1976).Google Scholar
7Kolmogorov, A. N.. On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval. Amer. Math. Soc. Transl. 4 (1949), 233–243.Google Scholar
8Landau, E.. Einige Ungleichungen für zweimal differentzierbare Funktionen. Proc. London Math. Soc. 13 (1913), 43–49.Google Scholar
9Rivlin, T. J.. The Chebyshev Polynomials (New York: Wiley, 1974).Google Scholar
10Schoenberg, I. J.. The elementary cases of Landau's problem of inequalities between derivatives. Amer. Math. Monthly 80 (1973), 121–158.CrossRefGoogle Scholar
11Vaserman, L. M. and Lepin, A. Ya.. Generalized Bernstein conditions for nonlinear systems of ordinary differential equations. Differentsial'nye Uravneniya 5 (1969), 1107–1113.Google Scholar
12Kwong, M. K. and Zettl, A.. Ramifications of Landau's inequality. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 175–212.CrossRefGoogle Scholar