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:
On the real lineEdit
For k = 1, n = 2, T=R the inequality was first proved by Edmund Landau 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:
where an are the Favard constants.
On the half-lineEdit
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.
- Weisstein, E.W. "Landau-Kolmogorov Constants". MathWorld--A Wolfram Web Resource.
- Landau, E. (1913). "Ungleichungen für zweimal differenzierbare Funktionen". Proc. London Math. Soc. 13: 43–49. doi:10.1112/plms/s2-13.1.43.
- 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.
- 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.
- 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.