Problem 225

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claims/: The 2 claim pages of Problem 225, one per claimant's result; the problem's standing derives from them.

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Statement. Let

$$f(\theta) = \sum_{0\leq k\leq n}c_k e^{ik\theta}$$

be a trigonometric polynomial all of whose roots are real, such that $\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1$. Then

$$\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.$$

Statement (corrected). Let

$$f(\theta) = \sum_{0\leq k\leq n}c_k e^{ik\theta}$$

with $n\geq1$ and $c_n\neq0$ be a trigonometric polynomial all of whose $n$ roots are real, such that $\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1$. Then

$$\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.$$

Notes. The site's display admits degenerate cases for which the bound fails. The constant $f\equiv1$ ($n=0$) has no roots, maximum $1$ and integral $2\pi$. If roots are read as zeros of $f$ as an entire function of $\theta$, $f(\theta)=e^{im\theta}$ fails the same way. The problem's sources state the conjecture with every root counted. Kristiansen states it for a trigonometric polynomial of degree $n\ge1$ with real coefficients and $2n$ real roots (Proc. Amer. Math. Soc. 44 (1974), p. 49). Saff and Sheil-Small state it for one of degree $n$ with all $2n$ zeros in $[0,2\pi)$ real (their Conjecture 1). The corrected Statement follows them: $n\geq1$ and $c_n\neq0$, so that $n$ is the actual degree of $P(z)=\sum_{k=0}^n c_kz^k$, and every one of the $n$ algebraic roots of $P$, counting multiplicity, on the unit circle, of the form $e^{i\theta}$ with $\theta$ real. The site's display does not state a root count; the full-root reading is the hypothesis used in the exact transfer below. The corrected Statement is proved, by Kristiansen [Kr74] (a two-sided real theorem, which the site describes as the real-coefficient case) and by Saff and Sheil-Small [SaSh74] in general (see Status). The site's wording fails at the degenerate cases above, and no other result about it is recorded.

Saff and Sheil-Small also prove an equivalent two-sided theorem. Their degree-$n$ trigonometric polynomial has $2n$ real zeros in a period and is converted to a degree-$2n$ algebraic polynomial. That normalization is recorded separately rather than identified with the one-sided display.

Status. Proved, the site's label. The site credits two independent solutions, Kristiansen [Kr74], which it describes as the real-coefficient case, and Saff and Sheil-Small [SaSh74] for general complex coefficients. The claim pages [[problems/analysis/E0225/claims/1974_11_01_saff_sheil_small|Saff and Sheil-Small 1974]], accepted on its refereed publication and the site's credit, and Kristiansen 1974, accepted on its refereed publication, record them. Kristiansen's theorem is the two-sided real form, a real trigonometric polynomial of degree $n\ge1$ with $2n$ real roots, which implies the display for arbitrary complex coefficients; the site's description is narrower than his paper. Both routes prove the corrected Statement; this project's own review of the compiled chain is described under Review record below and awards no standing.

Source. erdosproblems.com/225, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #225, https://www.erdosproblems.com/225.

References.

• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.
• [Kr74] Kristiansen, G. K., Proof of an inequality for trigonometric polynomials. Proc. Amer. Math. Soc. (1974), 49--57.
• [Kr76] Kristiansen, G. K., Erratum to ``Proof of a polynomial conjecture'' (Proc. Amer. Math. Soc. 44 (1974), 58--60). Proc. Amer. Math. Soc. (1976), 377. The site's commentary says that this erratum fixed an error in Kristiansen's proof of this problem; the erratum itself concerns the companion note Proof of a polynomial conjecture, on real polynomials with all roots in an interval, not [Kr74], and reports no gap in [Kr74]. The claim page records the account.
• [SaSh74] [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/_index|Saff, E. B. and Sheil-Small, T., Coefficient and integral mean estimates for algebraic and trigonometric polynomials with restricted zeros]]. J. London Math. Soc. (2) 9, no. 1 (November 1974), 16--22.

Formalization. The statement file ErdosProblems/225.lean of formal-conjectures, at its commit of 20 September 2026, declares erdos_225 under category research solved: for $n>0$ and coefficients $c$ with $c_0\neq0$ and $c_n\neq0$, if every zero of $f(z)=\sum_{k\le n}c_ke^{ikz}$ as an entire function of $z$ is real and $|f|\le1$ on $[0,2\pi]$ with the value $1$ attained, then $\int_0^{2\pi}|f|\le4$. Its docstring reads "all roots real" as the zeros of $f$ as an entire function of $\theta$, and normalizes by $c_0c_n\neq0$ and $n\geq1$. The condition $c_0\neq0$ is extra to the positive-degree convention above: the entire-function reading needs it, since a factor $e^{im\theta}$ has no zeros and changes neither the zeros nor $|f|$ on the real line, while under the convention above, where every zero of $P$ lies on the unit circle, $c_0=P(0)\neq0$ is automatic. The file carries a formal_proof attribute pointing at the Lean development in Boris Alexeev's lean-proofs repository, at the commit and anchor the attribute names; that development declares itself a formalization of Saff and Sheil-Small's solution, with Codex and GPT-5.6 Sol as its formal authors, and is linked and described on [[problems/analysis/E0225/claims/1974_11_01_saff_sheil_small|their claim page]]. The community database (teorth/erdosproblems,) records the statement formalized since 20 September 2026 and the status proved with no Lean qualification; the site's indicator reads "Formalised statement? Yes". This corpus has built and audited neither file, and no kernel credit is claimed.

Current assessment

The recorded proof proves the corrected Statement above. The standing rests on the two claim pages named in the Status: Saff and Sheil-Small's refereed paper settles the complex case, credited by the site, and Kristiansen's refereed paper proves the two-sided real form, which implies the display for arbitrary complex coefficients, though the site describes it as the real-coefficient case; the Kristiansen paper is not held in the library, and the account of the cited 1976 erratum is on his claim page. This project's own reviews of the compiled Theorem 1 chain, the Theorem 2 reduction and the transfer sentence are described under Review record below; none of them gives formal-verification or new acceptance credit. This page records no dated status-search scope. The 1976 erratum the site cites concerns Kristiansen's companion note Proof of a polynomial conjecture (Proc. Amer. Math. Soc. 44 (1974), 58--60), whose proof misses one case, and reports no gap in the trigonometric paper, which cites only Erdős's 1940 note; the Saff--Sheil-Small argument does not rest on either Kristiansen paper.

Progress

Let

$$P(z)=\sum_{k=0}^n c_kz^k.$$

Then $f(\theta)=P(e^{i\theta})$, and the full-root convention says exactly that all zeros of $P$ lie on the unit circle. The complete [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/theorem_1|Theorem 1]] gives, for every $q>0$,

$$\int_0^{2\pi}|P(e^{i\theta})|^q\,d\theta \leq A_q\left(\frac M2\right)^q, \qquad M=\max_{|z|=1}|P(z)|.$$

At $q=1$,

$$A_1=\int_0^{2\pi}|1+e^{i\theta}|\,d\theta =\int_0^{2\pi}2|\cos(\theta/2)|\,d\theta=8.$$

The problem assumes $M=1$, so the theorem gives

$$\int_0^{2\pi}|f(\theta)|\,d\theta \leq 8\left(\frac12\right)=4.$$

The paper's [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/theorem_2|Theorem 2]] proves the same $4M$ bound for the equivalent two-sided degree-$n$ setting with all $2n$ zeros real. Its proof reduces that setting to Theorem 1 through $T_n(\theta)=e^{-in\theta}P_{2n}(e^{i\theta})$.

The author-hosted copy of the paper, linked from the claim page, is a seven-page galley whose first-page footer carries provisional volume and page data; the final article is J. London Math. Soc. (2) 9, no. 1 (November 1974), 16--22. The result pages cite the galley's page numbers.

Known Results

• [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/theorem_1|Saff--Sheil-Small Theorem 1]]: the complete algebraic integral-mean proof, equality case, and exact one-sided $q=1$, $M=1$ transfer above.
• [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/theorem_2|Saff--Sheil-Small Theorem 2]]: the equivalent two-sided trigonometric result and equality case.

The references also record Kristiansen's proof and a 1976 correction. Kristiansen proves the conjecture for a real trigonometric polynomial of degree $n\ge1$ with $2n$ real roots, which gives the display for arbitrary complex coefficients; the site's description of it as the real-coefficient case is narrower than the paper. The correction concerns his companion note Proof of a polynomial conjecture (Proc. Amer. Math. Soc. 44 (1974), 58--60), not the trigonometric paper, as his claim page records; the Saff--Sheil-Small argument does not rest on either.

Review record. The [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/evidence/verify/full_proof_review|full-proof review]] independently checked the complete Theorem 1 reconstruction and its interfaces as the pages then stood, identified the positive-degree endpoint, and passed the Theorem 2 reduction and the exact transfer above conditionally on the literal endpoint corrections it requested. The [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/evidence/verify/publication_review|publication review]] of the corrected pages retains its PASS verdict as a disclosed non-blind delta review and warrants no independent acceptance of the corrected endpoint text; the [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/evidence/verify/publication_review_fresh|fresh blind review]] of the pages as they stood on 2026-09-18T07:24:04Z returned refutation-failed for Theorem 2 and the transfer, and its [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/evidence/verify/publication_review_grade_fresh|distinct grade]] records PASS for the report contract and void for independence. The third-round [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/evidence/verify/publication_review_r3|blind review]] of the pages as they stood on 2026-09-18T07:24:04Z, with its [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/evidence/verify/publication_grade_r3|distinct grade]] recording pass for the report contract and pass for independence, returned refutation-failed for Theorem 1 as consumed and for Theorem 2 under its positive-degree interpretation and found a defect in the Theorem 1 page's transfer sentence at $n=0$ (repair: carry $n\geq1$, or "$f$ nonconstant", in that section); the grade states that it warrants acceptance of the two theorems and no acceptance of the transfer sentence as it then stood. The transfer sentence was repaired on 2026-09-18, after the third-round review, and the focused [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/evidence/verify/transfer_repair_review|transfer repair review]] of the section as it stood on 2026-09-18T09:26:42Z returned refutation-failed for the repaired section with no defect; its [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/evidence/verify/transfer_repair_grade|distinct grade]] records pass for the report contract and pass for independence and states that the repaired section survives focused blind review, conditional on Theorem 1 as the card states it, with no tier asserted. The Theorem 1 page's transfer sentence is therefore the reviewed text, and the source card discloses the later rewording of 2026-10-07. The displayed problem Statement is unchanged; the corrected Statement and its Notes immediately below it record the required actual-degree convention. This record gives no formal-verification or new acceptance credit.

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.

• saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros
• saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros / theorem_1
• saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros / theorem_2