
> 
> **Question.** If $f(z)\in\mathbb{C}[z]$ is a [monic polynomial](https://en.wikipedia.org/wiki/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](https://www.math.temple.edu/~tewodros/Catalan_Convexity.pdf).



How small (in modulus) can a polynomial get?
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>


