$n$ points $z_1,z_2,\cdots,z_n$ in the unit open disk are given. Prove or disprove that there exists $z$ in the unit circle such that $\prod_{i=1}^n |z-z_i|^i \ge 1$.
I think it can be solved by elementary analytic methods but no idea.
Assume that $z_i \neq 0$ for all $i$ - think about why you can assume that. For positive multiplicities $m_i$, consider the polynomials
$$P(z) = \prod_{i = 1}^n (z- z_i)^{m_i}$$
and
$$Q(z) = z^{\sum_{i = 1}^n m_i}\cdot P(1/z).$$
How are the values of $P$ and $Q$ on the unit circle related, and what does the maximum modulus theorem say about the values of $Q$ on the unit circle?