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]
- [math]\displaystyle{ \|f^{(k)}\|_{L_\infty(T)} \le C(n, k, T) {\|f\|_{L_\infty(T)}}^{1-k/n} {\|f^{(n)}\|_{L_\infty(T)}}^{k/n} \text{ for } 1\le k \lt n. }[/math]
On the real line
For k = 1, n = 2 and T = [c,∞) or T = R, the inequality was first proved by Edmund Landau[2] with the sharp constants C(2, 1, [c,∞)) = 2 and C(2, 1, R) = √2. Following contributions by Jacques Hadamard and Georgiy Shilov, Andrey Kolmogorov found the sharp constants and arbitrary n, k:[3]
- [math]\displaystyle{ C(n, k, \mathbb R) = a_{n-k} a_n^{-1+k/n}~, }[/math]
where an are the Favard constants.
On the half-line
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
There are many generalisations, which are of the form
- [math]\displaystyle{ \|f^{(k)}\|_{L_q(T)} \le K \cdot {\|f\|^\alpha_{L_p(T)}} \cdot {\|f^{(n)}\|^{1-\alpha}_{L_r(T)}}\text{ for }1\le k \lt n. }[/math]
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
- ↑ Weisstein, E.W.. "Landau-Kolmogorov Constants". MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Landau-KolmogorovConstants.html.
- ↑ Landau, E. (1913). "Ungleichungen für zweimal differenzierbare Funktionen". Proc. London Math. Soc. 13: 43–49. doi:10.1112/plms/s2-13.1.43. https://zenodo.org/record/1447772.
- ↑ 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.
- ↑ Kallman, Robert R. (1970), "On the inequality [math]\displaystyle{ \Vert f^{\prime} \Vert^{2}\leqq4\Vert f\Vert\cdot\Vert f''\Vert }[/math]", Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), New York: Academic Press, pp. 187–192.
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