TITLE: Maximum of $|(z-a_1)\cdots(z-a_n)|$ on the unit circle
QUESTION [16 upvotes]: Let $a_1,\ldots,a_n$ be points on the unit circle. Let $P(z)=(z-a_1)\cdots(z-a_n)$. The maximum principle or Rouche's theorem can be used to show that there exists a point $b$ on the unit circle such that $|P(b)|\geq1$.
Question: What is the maximum value $c$ such that there always exists a point $b$ on the unit circle such that $|P(b)|\geq c$, regardless of what $a_1,\ldots,a_n$ are?

REPLY [3 votes]: In fact even more is true. Given any $n$ points $a_1,..a_n$ on the unit circle and weights $p_1,p_2...p_n\ge 0$ such that $p_1+p_2+...p_n=n$ one has
$$\max_{|z|=1}|z-a_1|^{p_1}|z-a_2|^{p_2}...|z-a_n|^{p_n}\ge 2.$$
 This was a conjecture from the paper by Vsevolod Lev and Sergey Konyagin from $2003$ and was proved in an extremely beautiful way by user "fedja" https://mathoverflow.net/questions/64099/the-maximum-of-a-polynomial-on-the-unit-circle
An alternative proof was given by Alexander Eremenko (see the same post), however, he used quite a powerful machinery (Dubinin theorem).