There is first Polya's estimate that if $f$ is a monic polynomial, then $$|\{x\in \mathbb{R}:\quad |f(x)|\leq 2\}| \leq 4.$$ A proof can be found in the book "Proofs from the book". One can obtain inequalities for non-monic polynomials by rescaling.
Let $f: G \to C$ \mathbb{C}$be analytic and assume that$f$is bounded by$1$on a disc of radius$2$. Then there are constant$C, c > 0$such that $$|\{z\in \mathbb{C}:\quad |z| < 1, | f(z)| \leq e^{-s}\}| \leq C \exp\left( - \frac{c}{\log(\varepsilon^{-1})} s \right)$$ where$\varepsilon = |f(0)|$. In fact, this is sharper, since it provides some information on how the set looks. For a polynomial it's just the union of its degree many disks. (For analytic functions countably many). 2 added 215 characters in body There is first Polya's estimate that if$f$is a monic polynomial, then $$|\{x:\quad \{x\in \mathbb{R}:\quad |f(x)|\leq 2\}| \leq 4.$$ A proof can be found in the book "Proofs from the book". One can obtain inequalities for non-monic polynomials by rescaling. Second there is Cartan's lemma or estimate. It can for example be found in Levin's book on entire functions. The estimate even holds for analytic functions. The basic statement is: Let$f: G \to C$be analytic and assume that$f$is bounded by$1$on a disc of radius$2$. Then there are constant$C, c > 0$such that $$|\{z:\quad \{z\in \mathbb{C}:\quad |z| < 1, | f(z)| \leq e^{-s}\}| \leq \exp\left( - \frac{c}{\log(\varepsilon^{-1})} s \right)$$ where$\varepsilon = |f(0)|$. In fact, this is sharper, since it provides some information on how the set looks. For a polynomial it's just the union of its degree many disks. (For analytic functions countably many). 1 There is first Polya's estimate that if$f$is a monic polynomial, then $$|\{x:\quad |f(x)|\leq 2\}| \leq 4.$$ A proof can be found in the book "Proofs from the book". One can obtain inequalities for non-monic polynomials by rescaling. Second there is Cartan's lemma or estimate. It can for example be found in Levin's book on entire functions. The estimate even holds for analytic functions. The basic statement is: Let$f: G \to C$be analytic and assume that$f$is bounded by$1$on a disc of radius$2$. Then there are constant$C, c > 0$such that $$|\{z:\quad |z| < 1, | f(z)| \leq e^{-s}\}| \leq \exp\left( - \frac{c}{\log(\varepsilon^{-1})} s \right)$$ where$\varepsilon = |f(0)|\$.