Problem 115

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claims/: The 1 claim page of Problem 115, one per claimant's result; the problem's standing derives from them.

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Statement. If $p(z)$ is a polynomial of degree $n$ such that ${z : \lvert p(z)\rvert\leq 1}$ is connected then is it true that

$$\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?$$

Statement (corrected). If $p(z)=z^n+\ldots$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected then is it true that

$$\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?$$

Notes. The site's wording puts no normalization on $p$. The change writes $p(z)=z^n+\ldots$, so that $p$ is monic; nothing else changes. The evidence is Erdős's own statement of the problem. [Er61], Problem IV.1, p. 246 (Some unsolved problems), opens "Let $z^n+\ldots$ be a polynomial of degree $n$" and asks on p. 247 whether $\max_{z\in E_f^{(n)}}\lvert f'(z)\rvert<\frac{n^2}{2}$ when $E_f^{(n)}$ is connected, his (IV.1.1). Hayman's collection states the question as Problem 4.8, one of five problems on the set $E_f^{(n)}$, the first of which, Problem 4.7, takes $f(z)=z^n+a_1z^{n-1}+\ldots+a_n$ (Research Problems in Function Theory, the 2018 edition, which keeps the 1967 numbering), and Eremenko and Lempert state it from there with $f(z)\sim z^n$ as $z\to\infty$, their (1) on p. 191. The site's own commentary fits only monic polynomials: it says the maximum is at least $n$, with equality only for $p(z)=z^n$, and for $cz^n$ with $\lvert c\rvert<1$ the maximum is $n\lvert c\rvert^{1/n}<n$. The defect is the site's; Erdős's text carries the normalization. The form comes from [Er61], not from the hypotheses of the theorem that settles it. No result about the site's wording is recorded.

Formulation. Erdős asked for the exact bound $n^2/2$ in place of $(\tfrac12+o(1))n^2$ [Er61, p. 247, (IV.1.1)], with a strict inequality, and added that (IV.1.1), if true, is best possible, as the $n$-th Chebyshev polynomial shows. That question has the answer no for every $n$: Eremenko and Lempert's extremal polynomial $f_n(z)=T_n(2^{(1-n)/n}z+1)$ is monic, has its critical values at $\pm1$, so that its set $E_{f_n}$ is connected, and has $f_n(0)=1$ and $f_n'(0)=2^{1/n-1}n^2>n^2/2$ [ErLe94, pp. 191--192]. The site records that Szabados observed that the bound without the $o(1)$ term is too strong; Eremenko and Lempert report that a later survey of Erdős noted that the bound must be replaced by $\tfrac12\{1+o(1)\}n^2$, and Hayman's Update 4.8 records Eremenko's remark that the bound $\tfrac12n^2$ fails for Chebyshev's polynomials. The site states the problem with the $o(1)$ term, and that question, made monic, sets the standing.

Status. Proved. The site's label is PROVED (LEAN), a label that describes the corrected Statement; its Lean marker refers to a formal proof by others, linked from the claim page, unbuilt and unaudited in this corpus. The corrected Statement is proved: Eremenko and Lempert [ErLe94] prove the sharp bound $2^{1/n-1}n^2$, attained by a shifted Chebyshev polynomial, refereed in the Proceedings of the AMS and credited by the site, which the corpus accepts on the claim page Eremenko and Lempert 1994.

Source. erdosproblems.com/115, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #115, https://www.erdosproblems.com/115.

References.

• [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221–254; Problem IV.1, pp. 246–247, which opens "Let $z^n+\ldots$ be a polynomial of degree $n$" and poses (IV.1.1), $\max\lvert f'(z)\rvert<n^2/2$ over a connected $E_f$.
• [ErLe94] Erëmenko, A. and Lempert, L., An extremal problem for polynomials. Proc. Amer. Math. Soc. (1994), 191-193.
• [Po59a] Pommerenke, Ch., On the derivative of a polynomial. Michigan Math. J. (1959), 373-375.

Formalization. Statement in formal-conjectures, marked solved there with a link to a Lean proof in the lean-proofs repository, which the claim page records at its pinned commit; this corpus has built and audited neither.

Current assessment

The corrected Statement asks whether a monic polynomial of degree $n$ with connected set $E_p=\{\lvert p(z)\rvert\le1\}$ has $\lvert p'\rvert\le(\tfrac12+o(1))n^2$ on $E_p$; the Notes give the evidence for the normalization, and the formal-conjectures statement also takes $p$ monic. The answer is yes: Eremenko and Lempert [ErLe94] prove $\max_{E_p}\lvert p'\rvert\le2^{1/n-1}n^2$, with equality only for rotations and translates of $T_n(2^{(1-n)/n}z+1)$, so the constant $\tfrac12$ is exact and the $o(1)$ term, which Szabados observed to be necessary, is $(2^{1/n}-1)/2$, about $(\log2)/(2n)$, since $2^{1/n-1}n^2=(\tfrac12+\tfrac12(2^{1/n}-1))n^2$. The lower bound $n$ is trivial, attained only by $z^n$, and the connectedness hypothesis cannot be dropped, as $z^2+10z+1$ shows; Pommerenke [Po59a] had the bound $\tfrac e2n^2$. The accepted claim page carries the acceptance (refereed, credited by the site) and records a Lean proof of the sharp bound by others. Proof coverage: the library card records Theorem 1 and its sharpness and does not verify the proof, so the paper's proof is unverified in this corpus, and the Lean development is unbuilt and unaudited.

Search scope: the site's problem page, the community database entry (proved with a Lean marker as of its last update), the formal-conjectures statement and the linked lean-proofs file at its pinned commit, and the library card Eremenko and Lempert 1994; no forum proof claim and no OpenAI release item names this problem. No wider literature search was made, none being needed for a refereed answer the site credits.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• eremenko_1994_extremal_problem_polynomials
• eremenko_1994_extremal_problem_polynomials / theorem_1
• hayman_lingham_2018_research_problems_function_theory
• hayman_lingham_2018_research_problems_function_theory / problem_4_8