In Appendix C3 of Shewchuk's excellent notes on conjugate gradient, it is stated without proof that
Chebyshev polynomials... increase in magnitude more quickly outside the range $[-1,1]$ than any other polynomial that is restricted to have magnitude no greater than one inside the range $[-1,1]$.
Here, we mean Chebyshev polynomials of the first kind, $T_n(x)=\cos(n \cos^{-1} x)$ for $x \in [-1,1]$. I have not been able to find a proof of this fact anywhere, and don't even know what I would search for in the first place. I have tried proving it myself, supposing that some polynomial attains a larger magnitude and attempting to arrive at a contradiction by showing, for example, that such a polynomial has too many zeros.
How might I begin to prove something like this / where can I find more information?