TITLE: How small (in modulus) can a polynomial get?
QUESTION [8 upvotes]: Question. If $f(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$, is it true that
  $$\max\{\,\vert f(x)\vert: \, -1\leq x\leq 1\}\geq 2^{1-n} \,\, ?$$

Context. This came up while working on some convexity problems.

REPLY [12 votes]: Yes. Proved by Chebyshev. The extreme case ($\max = 2^{1-n}$) is given by $2^{n-1}\cdot f(x)$ being a Chebyshev polynomial of first kind.
https://en.wikipedia.org/wiki/Chebyshev_polynomials

REPLY [5 votes]: You are essentially asking to approximate $x^n$ on $[-1,1]$ by a polynomial of degree less than $n$.  The Remez algorithm will find the best possible aproximation.  In Maple you can enter 
numapprox[minimax](x^n,x=-1..1,[n-1,0]);
If you take $n=10$ or so and work to 100 digits you will quickly convince yourself that the maximum error is precisely $2^{1-n}$.  It also looks as though the optimal approximation $f(x)$ has $2^{1-n}f(x)\in\mathbb{Z}[x]$.  One could probably find a formula for the coefficients experimentally with the help of OEIS and then work back to a proper proof.  But sadly I need to go and do less interesting things now.