Let $g(x)\colon [-2,2]\to \mathbb{R}$ be a continuous function. Let $f_n(x)$ be a polynomial of degree $n$ such that $\log |f_n(x)|\leqslant ng(x)$ for all $x\in [-2,2]$. Then the maximal possible leading coefficient of $f_n(x)$ is at most $e^{Cn+o(n)}$, where $$C:=\frac1{\pi}\int_{-2}^2\frac{g(x)}{\sqrt{4-x^2}}dx.$$ (Well, this bound works even without $o(n)$ term, that may be quickly proved as follows. We may assume that all roots of $f_n$ belong to $[-2,2]$, otherwise we may change one of quadratic or linear factors in the $f_n$ real factorization so that all its values on the segment $[-2,2]$ decrease. Now if $f_n(x)=e^{An}\prod_{i=1}^n(x-x_i)$, then $g(x)\geqslant A+\frac1n\sum \log|x-x_i|$, dividing by $\pi\sqrt{4-x^2}$ and integrating we get $C\geqslant A$, since $\int_{-2}^2\frac{\log|x-x_i|}{\sqrt{4-x^2}}dx=0$ whenever $x_i\in [-2,2]$.)
I am interested when this bound is tight, and when it is not, what is the tight bound.
