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)
\| | |
Linfty(T) |
\leC(n,k,T)
{\|f\| | |
Linfty(T) |
For k = 1, n = 2 and T = [''c'',∞) or ''T'' = '''R''', the inequality was first proved by Edmund Landau<ref>{{cite journal|first=E.|last= Landau|title=Ungleichungen für zweimal differenzierbare Funktionen|journal=Proc. London Math. Soc.|volume=13|year=1913|pages=43–49|doi=10.1112/plms/s2-13.1.43 |url= https://zenodo.org/record/1447772}}</ref> 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:[2]
C(n,k,R)=an-k
-1+k/n | |
a | |
n |
~,
where an are the Favard constants.
Following work by Matorin and others, the extremising functions were found by Isaac Jacob Schoenberg,[3] explicit forms for the sharp constants are however still unknown.
There are many generalisations, which are of the form
\|f(k)
\| | |
Lq(T) |
\leK ⋅
\alpha | |
{\|f\| | |
Lp(T) |
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.[4]
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