- Kolgomorov's inequality
Kolmogorov's inequality is an
inequality which gives a relation among a function and its first and secondderivative s. 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
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