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.