Landau–Kolmogorov inequality

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In mathematics, the Landau–Kolmogorov inequality, named after Edmund Landau and Andrey Kolmogorov, is the following family of interpolation inequalities between different derivatives of a function f defined on a subset T of the real numbers:[1]

On the real line[edit]

For k = 1, n = 2, T=R the inequality was first proved by Edmund Landau[2] with the sharp constant C(2, 1, R) = 2. Following contributions by Jacques Hadamard and Georgiy Shilov, Andrey Kolmogorov found the sharp constants and arbitrary n, k:[3]

where an are the Favard constants.

On the half-line[edit]

Following work by Matorin and others, the extremising functions were found by Isaac Jacob Schoenberg,[4] explicit forms for the sharp constants are however still unknown.

Generalisations[edit]

There are many generalisations, which are of the form

Here all three norms can be different from each other (from L1 to L, with p=q=r=∞ in the classical case) and T may be the real axis, semiaxis or a closed segment.

The Kallman–Rota inequality generalizes the Landau–Kolmogorov inequalities from the derivative operator to more general contractions on Banach spaces.[5]

Notes[edit]

  1. ^ Weisstein, E.W. "Landau-Kolmogorov Constants". MathWorld--A Wolfram Web Resource.
  2. ^ Landau, E. (1913). "Ungleichungen für zweimal differenzierbare Funktionen". Proc. London Math. Soc. 13: 43&ndash, 49. doi:10.1112/plms/s2-13.1.43.
  3. ^ Kolmogorov, A. (1962). "On Inequalities Between the Upper Bounds of the Successive Derivatives of an Arbitrary Function on an Infinite Integral". Amer. Math. Soc. Transl. 1–2: 233&ndash, 243.
  4. ^ Schoenberg, I.J. (1973). "The Elementary Case of Landau's Problem of Inequalities Between Derivatives". Amer. Math. Monthly. 80: 121&ndash, 158. doi:10.2307/2318373.
  5. ^ Kallman, Robert R.; Rota, Gian-Carlo (1970), "On the inequality ", Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), New York: Academic Press, pp. 187–192, MR 0278059.