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Erdős problem 1039

For f(z)=∏i=1n(z−zi)f(z) = \prod_{i=1}^n (z - z_i) with all ∣zi∣≤1|z_i| \le 1, let ρ(f)\rho(f) be the radius of the largest disc contained in {z:∣f(z)∣<1}\{z : |f(z)| < 1\}. Is ρ(f)≫1/n\rho(f) \gg 1/n? The worst case is now known to be Θ(1/n)\Theta(1/n), with the explicit bound ρ(f)≥(log⁡2)/n\rho(f) \ge (\log 2)/n.

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