$f(x) \in C[-1,1]$ is a polynomial of degree n,and $|f(x)|\leqslant 1$,Prove:$|f'(x)|\leqslant n^2$

Asked Dec 31 '15 at 13:04

Active Dec 31 '15 at 13:25

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I met this problem ,But i don't know how to do it.

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$

and $f'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+\cdots+a_1$

But how to do next? Who can help me ! Thanks!

edited Dec 31 '15 at 13:25

A.Γ.

asked Dec 31 '15 at 13:04

smallsmallice

5

"Markov brothers' inequality" might be a good keyword. – Clement C. Dec 31 '15 at 13:13
1

Twelve Proofs of the Markov Inequality – Martin R Dec 31 '15 at 13:22
@MartinR Did you mean this question? – A.Γ. Dec 31 '15 at 13:46
@A.G.: It seems that I confused things, I had used the inequality in an answer. – Yes, that is more or less the same question, and the answer has at least a link to a proof. I don't know if that counts as a duplicate if this question asks for a proof explicitly. – Martin R Dec 31 '15 at 13:56
This is a consequence of the Markov-Bernstein inequality. See this question: http://math.stackexchange.com/questions/673367/polynomial-boundaries/673659#673659 The answer doesn't give a proof, but it tells you where to find twelve different proofs. – bubba Dec 31 '15 at 15:26

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