A kind of isoperimetric inequality for polynomials?

Question

During my programming of an app I stumbled upon the following question:

Suppose you are given a monic polynomial $f \in \mathbb{C}[x]$. Consider $f$ as a function and let $D \subset \mathbb{C}$ be the unit disc in the target complex plane. Then we can compute the volume $V_f := \int_{f^{-1}(D)} 1 dx$ of the preimage of $D$ with respect to the Lebesgue measure $dx$ on the source.

Computational experiments yield that there is an upper bound on $V_f$ as $f$ ranges over all (Edit) monic polynomials. My guess is that the maximum is achieved whenever $f$ has exactly one root of multiplicity $n = \operatorname{deg} f$. To me, this is very similar to the isoperimetric inequality in the sense that we are looking for an "optimal shape" determined by the polynomial $f$ in order to maximize a volume.

Yet, I do not know of any mathematics that treat this or a related question. Do you?

Answer 1

The following result is quoted in Edward Crane, The Area of Polynomial Images and Preimages, confirming your conjecture:

Theorem 1 (Pólya's inequality). Let $p$ be a monic polynomial of degree $n$ over $\Bbb C$ and let $D$ be a disc in $\Bbb C$ . Then the Euclidean area of $p^{-1}(D)$ is at most $$ \text{Area}(p^{-1}(D) \le \pi \left( \frac{\text{Area}(D)}{\pi} \right)^{1/n} \, , $$ with equality only when $p : z \to a(z − b)^n + c$ and the center of $D$ is $c$, the unique critical value of $p$.