PROVED
This has been solved in the affirmative.
If $p(z)$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected then is it true that\[\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?\]
The lower bound is easy: this is $\geq n$ and equality holds if and only if $p(z)=z^n$. The assumption that the set is connected is necessary, as witnessed for example by $p(z)=z^2+10z+1$.
The
Chebyshev polynomials show that $n^2/2$ is best possible here. Erdős originally conjectured this without the $o(1)$ term but Szabados observed that was too strong. Pommerenke
[Po59a] proved an upper bound of $\frac{e}{2}n^2$.
Eremenko and Lempert
[ErLe94] have shown this is true, and in fact Chebyshev polynomials are the extreme examples.
View the LaTeX source
Additional thanks to: Stefan Steinerberger
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #115, https://www.erdosproblems.com/115, accessed 2026-01-16
