Size of near-zero set of polynomial Let $p(x)$ be a polynomial of degree $D$ in one variable.  I am interested in the set
$$
A_{\varepsilon} := \{x\in \mathbb{R} \,\mid\, |p(x)|\leq \varepsilon |p'(x)|\}.
$$
Here $p'(x)$ denotes the derivative of $p$.  I wonder if there is there is a nice bound for the size of $A_\varepsilon$.  In particular, my question is:

Does there exists a constant $C_D$ (which is allowed to depend on the
  degree $D$) such that  $$ \mu(A_\varepsilon) \leq C_D \varepsilon. $$

Some examples:
I think that the case that $p(x)=x^D$ is pretty interesting.  In this case, the polynomial has a root of order $k$ at the origin, and 
$$
A_\varepsilon = [-\varepsilon D, \varepsilon D].
$$
This makes me think that perhaps one could take $C_D = 2D$.  
This examples also suggests that perhaps $A_\varepsilon$ is actually contained near the roots of the polynomials.  This can't be quite true, at least over $\mathbb{R}$, because of the example $p(x)=x^2+\varepsilon^{4}$.  This polynomial has no real roots, but it behaves just like $x^2$ for the purposes of this problem. 

I figured it out, albeit with a suboptimal constant $C_D = D^2$.
The first observation is that $x\in A_\varepsilon$ if and only if
$|p'(x)|/|p(x)| \geq \varepsilon^{-1}$.  This is a simple rearrangement, but the point is that there is a nice formula for $p'(x)/p(x)$.  To see this, factor $p$ over the complex numbers.  If $\{r_i\}_{i=1}^D$ is the set of complex roots, we can write $p(z) = c \prod_{i=1}^D (z-r_i)$.  Then 
$$
\frac{p'}{p} = \frac{d}{dz} \log p = \sum_{i=1}^D \frac{1}{z-r_i}.
$$
Now if the magnitude of the sum is larger than $\varepsilon^{-1}$, it must be that at least one of the terms has magnitude at least $(D\varepsilon)^{-1}$.
Thus we observe
$$
A_\varepsilon \subset \bigcup_{i=1}^D B_{D\varepsilon}(r_i)
$$
where $B_{D\varepsilon}$ denotes a ball of radius $D\varepsilon$, and we think of both sets as being subsets of the complex plane.  From this we conclude 
$$
|A_\varepsilon| \leq 2D^2\varepsilon.
$$
This is likely suboptimal, because we crudely estimated the sum of the reciprocals.  I would still be interested to learn what the best possible $C_D$ is.
