Landau-Kolmogorov inequality

In mathematics, the Landau-Kolmogorov inequality is an inequality between different derivatives of a function. There are many inequalities holding this name (sometimes they are also called Kolmogorov type inequalities), common formula is

: $|f^\{(k)\}|\_\{L\_q(T)\}\; le\; K\; cdot\; |f|^alpha\_\{L\_p(T)\}\; cdot\; |f^\{(n)\}|^\{1-alpha\}\_\{L\_r(T)\}$, where

Here all three norms can be different from each other (from $scriptstyle\; L\_1$ to $scriptstyle\; L\_infty$), giving different inequalities. These inequalities also can be written for function spaces on axis, semiaxis or closed segment (it is denoted by "T")—it also gives bunch of different inequalities. These inequalities are still intensively studied. Most honourable results are those where exact value of minimal constant $scriptstyle\; K=K\_T(n,k,q,p,r)$ is found.

History

An inequality of this type was firstly established by Hardy, Littlewood and Pólya. Some exact constants ($scriptstyle\; K\_\{R\_+\}(2,\; 1,\; infty,\; infty,\; infty)\; =\; 2$) were found by Landau in

: E. Landau, “Ungleichungen für zweimal differenzierbare Funktionen”, Proc. London Math. Soc., 13 (1913), 43–49.

Kolmogorov obtained one of the most outstanding results in this field, with all three norms equal to $scriptstyle\; L\_infty$, he found $scriptstyle\; K\_\{R\}(n,\; k,\; infty,\; infty,\; infty)$.