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]

\|f^{(k)}\|_{L_{\infty }(T)}\leq C(n,k,T){\|f\|_{L_{\infty }(T)}}^{1-k/n}{\|f^{(n)}\|_{L_{\infty }(T)}}^{k/n}{\text{ for }}1\leq k<n.

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]

C(n,k,\mathbb {R} )=a_{n-k}a_{n}^{-1+k/n}~,

where a_n 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

\|f^{(k)}\|_{L_{q}(T)}\leq K\cdot {\|f\|_{L_{p}(T)}^{\alpha }}\cdot {\|f^{(n)}\|_{L_{r}(T)}^{1-\alpha }}{\text{ for }}1\leq k<n.

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.
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 $\Vert f^{\prime }\Vert ^{2}\leqq 4\Vert f\Vert \cdot \Vert f''\Vert$ ", Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), New York: Academic Press, pp. 187–192, MR 0278059.

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[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 $\Vert f^{\prime }\Vert ^{2}\leqq 4\Vert f\Vert \cdot \Vert f''\Vert$ ", Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), New York: Academic Press, pp. 187–192, MR 0278059.