Kolgomorov's inequality

Kolmogorov's inequality is an inequality which gives a relation among a function and its first and second derivatives. Kolmogorov's inequality states the following:

Let $f\; colon\; mathbb\{R\}\; ightarrow\; mathbb\{R\}$ be a twice differentiable function on $mathbb\{R\}$ such that $f,$ and $f"\; ,$ are bounded on $mathbb\{R\}$. Denote

: $M\_0\; =\; sup\_\{xinmathbb\{R\; |f(x)|,\; M\_1\; =\; sup\_\{xinmathbb\{R\; |f\text{'}(x)|,\; M\_2\; =\; sup\_\{xinmathbb\{R\; |f"(x)|.$

Then, $f\text{'}\; ,!$ is bounded on $mathbb\{R\}$ and $M\_1\; le\; sqrt\{2M\_0M\_2\}$.

Proof

The proof of this inequality uses Taylor's theorem.

Let $a\; in\; mathbb\{R\}\_+^*,\; x\; in\; mathbb\{R\}$. Apply the Taylor-Lagrange Inequality to $f\; ,!$ on the intervals $[x-a,x]\; ,!$ and $[x,x+a]\; ,!$ and obtain

:
f(x-a)-(f(x)-af'(x))| le frac{a^2}{2}M_2\
f(x+a)-(f(x)+af'(x))| le frac{a^2}{2}M_2.end{cases} from which

::$|f(x+a)-f(x-a)-2af\text{'}(x)|\; ,!$:

so that

:$|2af\text{'}(x)|\; le\; |f(x+a)-f(x-a)|+a^2M\_2\; le\; 2M\_0+a^2M\_2.$

Hence,

: $M\_1\; le\; frac\{M\_0\}\{a\}+frac\{1\}\{2\}aM\_2\; le\; sqrt\{2M\_0M\_2\},$

where we have used the AM-GM inequality in the last step.

References

*cite book | author=Serge Francinou, Hervé Gianella, Serge Nicolas| title=Exercices de Mathématiques Oraux X-ENS| publisher=Cassini, Paris | year=2003 | id=ISBN 2-8425-032-X