Real functions Classical inequalities Theorem

Landau–Kolmogorov inequality

Statement

Let $n \ge 2$ and $1 \le k < n$ be integers. There is a constant $C(n, k)$ , depending only on $n$ and $k$ , with the following property. Let $f \colon \R \to \R$ be $n$ times differentiable, with $f$ and its $n$ -th derivative $f^{(n)}$ bounded on $\R$ . Then $f^{(k)}$ is bounded on $\R$ and $\norm{f^{(k)}}_\infty \le C(n, k) \, \norm{f}_\infty^{1 - k/n} \, \norm{f^{(n)}}_\infty^{k/n},$ where $\norm{g}_\infty = \sup_{x \in \R} \abs{g(x)}$ is the supremum of $\abs{g}$ .

Remarks

Proof idea for $n = 2$ and $k = 1$ : let $M_0 = \norm{f}_\infty$ and $M_2 = \norm{f''}_\infty$ . By Taylor's theorem, $f(x \pm h) = f(x) \pm h f'(x) + \frac{h^2}{2} f''(\xi_\pm)$ for some $\xi_\pm$ . Subtracting gives $\abs{f'(x)} \le \frac{M_0}{h} + \frac{h M_2}{2}$ for every $h > 0$ , and the choice $h = \sqrt{2 M_0 / M_2}$ gives $\norm{f'}_\infty^2 \le 2 M_0 M_2$ . The constant $2$ here is the best possible on $\R$ . On the half-line $[0, \infty)$ the same kind of bound holds with the best constant $4$ in place of $2$ .

The exponents are forced by scaling: replacing $f(x)$ by $f(\lambda x)$ multiplies $\norm{f^{(j)}}_\infty$ by $\lambda^j$ .

Going deeper

For a student

These are interpolation inequalities: the middle derivative is controlled by a weighted geometric mean of the two ends. They are a model for the Gagliardo–Nirenberg inequalities for Sobolev norms. The best constants $C(n, k)$ on $\R$ are known for all $n$ and $k$ , and they are attained by functions related to Euler splines. For $n = 2$ and $k = 1$ the best constant is $\sqrt{2}$ .

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Limit point of a set definition
Set bounded above definition
Limit of a function definition
Supremum definition
Derivative definition
Differentiable function definition
Derivative function definition
Higher derivative definition
Landau–Kolmogorov inequality statement

Tag	`C3DD4Z4MD1`
URI	`m2://C3DD4Z4MD1`
Kind	statement, theorem
Writing	W0 stub
MSC	26D10
Status	This copy of the site has no build data. Run `m2 status C3DD4Z4MD1`.
About	Higher derivative
Uses	Higher derivative, Supremum
Same as	Wikidata Q5638668 Wikipedia Landau–Kolmogorov inequality
Changed	2 November 2026