I'm in the process of reading a paper and I believe there's a mistake, but it could also be me not noticing something.
Here's the deal:
Let $\bar\Pi_k$ be the set of all polynomials of degree at most $k$ that satisfy $p(1) = 1$.
Next, let $\mu_i, i = 1...n$ be a finite set of numbers in the open interval $(-1,1)$.
In one of the proofs in the paper, they appear to make the following assumption: If $p \in \bar\Pi$, then $$ \max_i\, p^2(\mu_i) \le p^2(\max_i\, | \mu_i |)$$
However, I find absolutely no reason for this to be true. Note: The coefficients of the polynomials are not guaranteed to be non-negative, otherwise s.th. like Jensen's inequality might have done the trick.