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'(x)|, M_2 = sup_{xinmathbb{R |f"(x)|.

Then, $$f' ,! is bounded on $$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

:$$egin{cases}
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

::$$|f(x+a)-f(x-a)-2af'(x)| ,!:$$egin{alignat}{2}&=|(f(x+a)-(f(x)+af'(x)))-(f(x-a)-(f(x)-af'(x)))|\&le a^2M_2,\end{alignat}

so that

:$$|2af'(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