# bound on the leading coefficient of a polynomial

Given a polynomial with real coefficients, that satisfies $\forall x \in \left[ -1,1 \right]: \ |p(x)|\leqslant 1$, I have to show, that its leading coefficient, $a_m$ satisfies $a_m \leqslant 2^{m-1}$.

I have obtained some easy partial results: for example under the given conditions the sum of all even coefficients has to be in $\left[ -1,1 \right]$ (and the sum of the odd in $\left[ -2,2 \right]$), but that didn't help me much...

-

This is a property of Chebyshev polynomials. See the section "Minimal $\infty$-norm" there, where there's also a proof.