Let $p(x)$ be a polynomial of degree $n$, and let $\{z_i\}_{i=1}^n\subset\mathbb{C}$ be the set of all the zeros in the complex plane. Suppose also that $|z_i| \leq 5$ for each zero. Let $$ Z = \bigcup_{i=1}^n B_{(100n)^{-1}}(z_i) $$ be the set of all points that are within $(100n)^{-1}$ of a zero in the complex plane.
I am interested in the $L^2$ norm of $p$ on the interval $[-1,1]$, and based on a few examples I believe that the inequality $$ \int_{[-1,1]} |p(x)|^2 \,dx \leq 2 \int_{[-1,1] \setminus Z} |p(x)|^2\,dx $$ should hold. That is, I believe that it is impossible for most of the mass of the polynomial $p$ to be concentrated in the set $Z$.
Is this true? How can it be proven?
An interesting example is the case $p(x) = x^{n-1}(1-x)$. In this example most of the mass of the polynomial is within a ball of radius $10n^{-1}$ around the zero at $1$, but it is not too tightly concentrated near $1$.