Coefficient and Integral Mean Estimates for Algebraic and Trigonometric Polynomials with Restricted Zeros

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evidence/: Retains the independent full-proof review, the endpoint and publication review of the Theorem 1 chain and Problem 225 transfer, the later blind reviews with their grades.

external_inputs: Records the exact Lax, Gauss--Lucas, subordination, and beta-integral statements used in the Saff--Sheil-Small integral-mean theorem.

theorem_1: Proves the sharp L to the q bound for a polynomial all of whose zeros lie on the unit circle, including the equality characterization.

theorem_2: Transfers Theorem 1 to a two-sided trigonometric polynomial with all 2n zeros real and determines every equality case.

theorem_3: For a degree-n polynomial with all zeros on the unit circle and maximum modulus M there, every coefficient other than a middle one has modulus at most M/2, with equality only at the end coefficients of M(lambda z^n+mu)/2.

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E. B. Saff and T. Sheil-Small, Coefficient and Integral Mean Estimates for Algebraic and Trigonometric Polynomials with Restricted Zeros, Journal of the London Mathematical Society, second series 9, no. 1 (November 1974), 16--22. DOI.

The copy read for this card is the seven-page copy hosted on E. B. Saff's Vanderbilt page. It is an author-hosted galley/scan rather than a verified copy of the publisher's final PDF. Its pages 2--7 are numbered 002--007 (the first page prints no number), its production header says LMS JNL—53128—Saff—7pp, and the first-page footer still gives provisional volume and page text, [J. London Math. Soc. (2) 7 (1974) 000–000]. The publisher record separately identifies the published article as volume 9, issue 1, November 1974, pp. 16--22. Result locators below therefore use the physical/galley pages actually inspected and do not transfer final-page labels onto that copy. That copy is the London Mathematical Society's page proof from the author's site (https://math.vanderbilt.edu/saffeb/texts/16.pdf), a scan without text layer that prints no copyright line and states no terms; the publisher's page for the published article could not be read on 2026-10-02 (https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/s2-9.1.16 returned HTTP 403), and its Crossref record lists only the publisher's text-and-data-mining and terms-and-conditions links, which govern the published article and not this copy; the term is unstated.

Author-hosted source: https://math.vanderbilt.edu/saffeb/texts/16.pdf. Publisher record: https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-9.1.16. That copy is 317184 bytes.

Compiled scope

The complete selected chain is on physical pp. 1--3:

• [[analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/theorem_1|Theorem 1]] proves the sharp $L^q$ estimate for a degree-$n$ algebraic polynomial whose $n$ zeros lie on the unit circle. Its proof includes the self-inversive coefficient relation, the reflected derivative $Q$, identity (6), the finite Blaschke product $w$, pointwise estimate (7), the subordination step, and the equality case.
• [[analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/theorem_2|Theorem 2]] applies Theorem 1 to a two-sided trigonometric polynomial of degree $n$ having all $2n$ zeros real. It also determines the equality cases.
• [[analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/external_inputs|External inputs]] records the exact Lax, Gauss--Lucas, and subordination interfaces used by Theorem 1. Their proofs belong to other sources and are not recursively reconstructed here.
• [[analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/theorem_3|Theorem 3]] (galley p. 003), the last theorem of the paper's Section 2, bounds every coefficient of such a polynomial except a middle one by $M/2$. This is the paper's Conjecture 2, which it attributes to W. K. Hayman, with the middle coefficient of an even-degree polynomial excluded. It is recorded at claims-checked depth only and is outside the reviewed chain.

Theorem 1 is the literal route to the one-sided display on Problem 225. For $f(\theta)=\sum_{k=0}^n c_ke^{ik\theta}$, put $P(z)=\sum_{k=0}^n c_kz^k$. For $n\geq1$ and $c_n\neq0$, as Theorem 1 requires, the intended full-root convention puts all $n$ roots of $P$, counting multiplicity, on the unit circle. Theorem 1 with $q=1$ gives

$$\int_0^{2\pi}|f(\theta)|\,d\theta \leq A_1\frac{M}{2}=4M,$$

where $M=\max_{|z|=1}|P(z)|$ and $A_1=8$. Theorem 2 proves the two-sided form of the bound, but it uses the paper's different normalization: a degree-$n$ trigonometric polynomial with $2n$ real zeros and an associated algebraic polynomial of degree $2n$. Those two formulations are kept distinct.

Theorems 4--7 (the paper's Section 3, "Related results and conjectures"), Conjectures 3 and 4, Lemmas 1 and 2, and their proofs on physical pp. 3--7 are outside this compilation. No result or proof credit is claimed for them.

Historical route retained separately

The problem page also cites G. K. Kristiansen's real-coefficient proof and a 1976 correction. Neither the 1974 proof nor the correction was read for this card, and their exact article relationship and proof content are not recorded here. This Saff--Sheil-Small chain does not resolve that bibliographic gap and does not present the Kristiansen route as reviewed.

Bears on. Problem 225: Theorem 1 with $q=1$ and $A_1=8$ gives $\int_0^{2\pi}|f(\theta)|\,d\theta\le4$ for the problem's one-sided $f$ with maximum $1$, under the full-root reading with $n\ge1$ and $c_n\neq0$ (all $n$ roots of $\sum_k c_kz^k$ on the unit circle); Theorem 2 with $q=1$ proves the two-sided form, which the paper states as its Conjecture 1 and attributes to Erdős, for a trigonometric polynomial of degree $n\ge1$ with all its zeros real. The problem page records the standing.

Review record. An independent strong mathematical and source review verified Theorem 1 and every stated external interface at the frozen bytes, and passed the positive-degree Theorem 2 reduction and the exact Problem 225 transfer conditionally on the literal endpoint corrections it requested; the full-proof review retains that report, and the result pages make the reviewed endpoint restrictions explicit. The publication review of the corrected successor retains its PASS verdict but was ruled on 2026-09-18 a disclosed non-blind delta review, because its read set included the first review's verdict and pre-written review notices on the reviewed pages, so it warrants no independent acceptance of the successor's endpoint text. The fresh blind review of Theorem 2 and the transfer as they stood on 2026-09-18T07:24:04Z returned refutation-failed, and its distinct grade records PASS for the report contract and void for independence, because the commission routed the reviewer through prior-verdict text; the successor's endpoint text in theorem_2.md therefore remains author-supplied, with independent acceptance outstanding. The third-round blind review of 2026-09-18 read theorem_2.md, theorem_1.md, external_inputs.md and the Problem 225 Statement paragraph as they stood on 2026-09-18T07:24:04Z through a redacted extraction of those pages, and returned refutation-failed for Theorem 1 as consumed and for Theorem 2 under its positive-degree interpretation, and found a defect in the transfer sentence "Thus Theorem 1 applies with $M=1$" of theorem_1.md at the endpoint $n=0$ (witness $f\equiv1$, integral $2\pi>4$; correct for $n\geq1$; repair one clause). Its distinct grade records pass for the report contract and pass for independence, and states that it warrants acceptance of the two theorems and no acceptance of the transfer sentence as frozen; that sentence was repaired on 2026-09-18, after the review, so theorem_1.md's transfer section now carries $n\geq1$. The focused transfer repair review of 2026-09-18 read the Theorem 1 Statement, the repaired transfer section and the Problem 225 Statement paragraph as they stood on 2026-09-18T09:26:42Z through a redacted extraction of those pages and returned refutation-failed with no defect; its 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 transfer sentence is therefore accepted as the card states it. On 2026-10-07, after these reviews, the reviewed pages were reworded without changing any statement, hypothesis or proof step: the Theorem 1 Statement marks $n\geq1$ as implicit in the print, the Theorem 2 Statement attributes the multiplicity count to the card rather than the paper, Lax's inequality and both equality clauses are restated, and the Theorem 2 page and the transfer section's closing pointer to it no longer call the two-sided theorem stronger or broader, since each theorem implies the other. The named external proofs, Kristiansen route, formal verification, and acceptance evidence remain outside this compilation.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.