Landau–Kolmogorov inequality

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 lineEdit

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-lineEdit

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.


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]


  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: 43–49. doi:10.1112/plms/s2-13.1.43.
  3. ^ Kolmogorov, A. (1949). "On Inequalities Between the Upper Bounds of the Successive Derivatives of an Arbitrary Function on an Infinite Interval". Amer. Math. Soc. Transl. 1–2: 233–243.
  4. ^ Schoenberg, I.J. (1973). "The Elementary Case of Landau's Problem of Inequalities Between Derivatives". Amer. Math. Monthly. 80 (2): 121–158. doi:10.2307/2318373. JSTOR 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.