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Erdős #1039 · analysis / polynomials

Let f(z)=∏i=1n(z−zi)∈C[z]f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z] with ∣zi∣≤1\lvert z_i\rvert \leq 1 for all ii. Let ρ(f)\rho(f) be the radius of the largest disc which is contained in {z:∣f(z)∣<1}\{z: \lvert f(z)\rvert< 1\}.

Determine the behaviour …

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let f(z)=∏i=1n(z−zi)∈C[z]f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z] with ∣zi∣≤1\lvert z_i\rvert \leq 1 for all ii. Let ρ(f)\rho(f) be the radius of the largest disc which is contained in {z:∣f(z)∣<1}\{z: \lvert f(z)\rvert< 1\}.

Determine the behaviour of ρ(f)\rho(f). In particular, is it always true that ρ(f)≫1/n\rho(f)\gg 1/n?

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Background

A problem of Erdős, Herzog, and Piranian, who note that f(z)=zn−1f(z)=z^n-1 has ρ(f)≤π/2n\rho(f) \leq \frac{\pi/2}{n}.

Pommerenke [Po61] proved that

ρ(f)≥12en2.\rho(f) \geq \frac{1}{2en^2}.

Krishnapur, Lundberg, and Ramachandran [KLR25] proved

ρ(f)≫1nlog⁡n.\rho(f) \gg \frac{1}{n\sqrt{\log n}}.

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1039 link

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