Erdős Problem #115 — Derivative Bounds for Polynomials with Connected Sublevel Sets

Erdős conjectured that for a degree-$n$ polynomial with connected unit lemniscate, the maximum of $|p'|$ on the unit disk is at most $(1/2+o(1))n^2$. The problem is now settled asymptotically; Chebyshev polynomials show that the constant $1/2$ cannot be improved.

Known results

• Szabados observed that Erdős's stronger bound with $1/2$ and no $o(1)$ term is false.
• Pommerenke proved the upper bound $(e/2)n^2$.
• Eremenko and Lempert proved the asymptotic bound $(1/2+o(1))n^2$ in 1994.
• Equality in the basic lower bound $\max|p'|\ge n$ holds only for $p(z)=z^n$.

Eremenko–Lempert resolution (1994)

Eremenko and Lempert established the conjectured asymptotic estimate, resolving the stated problem. No newer counterexample, competing claim, or AI-attributed result was found.

Current status (as of June 2026): The asymptotic assertion is proved, with the sharp leading constant $1/2$; the stronger exact bound without $o(1)$ is false.