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: ‖ f ( k ) ‖ L ∞ ( T ) ≤ C ( n , k , T ) ‖ f ‖ L ∞ ( T ) 1 − k / n ‖ f ( n ) ‖ L ∞ ( T ) k / n for 1 ≤ k < n . {\displaystyle \|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 with the sharp constants C(2, 1, [c,∞)) = 2 and C(2, 1, R) = √2.

Source: Wikipedia — Landau–Kolmogorov inequality (CC BY-SA 4.0)

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: ‖ f ( k ) ‖ L ∞ ( T ) ≤ C ( n , k , T ) ‖ f ‖ L ∞ ( T ) 1 − k / n ‖ f ( n ) ‖ L ∞ ( T ) k / n for 1 ≤ k < n . {\displaystyle \|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 with the sharp constants C(2, 1, [c,∞)) = 2 and C(2, 1, R) = √2.

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Source: Wikipedia "Landau–Kolmogorov inequality" · CC BY-SA 4.0

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