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Saff and Sheil-Small prove, as their Theorem 1, that a polynomial $P$ of degree $n\geq1$ all of whose $n$ zeros lie on the unit circle satisfies, for every real $q>0$,

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

with equality exactly for $P(z)=\frac M2(\lambda z^n+\mu)$, $|\lambda|=|\mu|=1$. Write $P(z)=\sum_{k=0}^nc_kz^k$ for the coefficients of the trigonometric polynomial $f$ of Problem 225, so that $f(\theta)=P(e^{i\theta})$. The hypothesis that every root of $f$ is real says, in the corrected Statement of the problem page, that every one of the $n$ zeros of $P$ lies on the unit circle. At $q=1$ the constant is $A_1=\int_0^{2\pi}2|\cos(\theta/2)|\,d\theta=8$, and $M=1$ is the problem's normalization, so the theorem gives

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

This proves the statement for arbitrary complex coefficients. The convention matters at two points. The theorem needs $n\geq1$ and $c_n\neq0$, so that $n$ is the actual degree: the constant $f\equiv1$ has maximum $1$ and integral $2\pi>4$, so the bound fails in degree $0$. And the roots are read as the $n$ roots of $P$, all on the unit circle, which forces $c_0\neq0$: read instead as zeros of $f$ as an entire function of $\theta$, the hypothesis admits $f(\theta)=e^{i\theta}$, which has no zeros, maximum $1$ and integral $2\pi>4$. The paper's Theorem 2 proves the same bound $4M$ for a trigonometric polynomial of degree $n$ with all $2n$ zeros in a period real, an equivalent two-sided normalization that the problem page records separately.

The library card [[../library/analysis/saff_sheil_small_1974_coefficient_integral_mean_estimates_restricted_zeros/_index|Saff and Sheil-Small 1974]] compiles Theorem 1 and Theorem 2 and the review record of the compiled chain. That record is this project's own work and is not acceptance evidence for this claim; the problem page's Review record describes it.

Formalization. The Lean file src/latest/ErdosProblems/Erdos225.lean of Boris Alexeev's lean-proofs repository, linked above at a pinned commit, declares itself a formalization of a solution to Problem 225 and names E. B. Saff and T. Sheil-Small as its informal authors and Codex and GPT-5.6 Sol as its formal authors. It is therefore recorded on this page as a formalization of this claim, not as an independent result. Its theorem erdos_225 takes $n>0$, $c_n\neq0$, $c_0\neq0$, every root of $P(z)=\sum_kc_kz^k$ on the unit circle, and the maximum of $|f|$ on $[0,2\pi]$ equal to $1$ (the bound everywhere and an angle attaining it), and concludes $\int_0^{2\pi}|f|\le4$; its corollary erdos_225_of_onlyRealAngularRoots, the form the formal-conjectures statement file points to, replaces the root hypothesis by the reality of every zero of $f$ as an entire function of $\theta$. The file contains no sorry and no axiom command at the pinned commit and imports two other modules of the same repository. This project has not built the file or audited its statement against the problem, so no formalized evidence is listed.

Acceptance. The paper is refereed: E. B. Saff and T. Sheil-Small, 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. The site's curator, Thomas F. Bloom, marks Problem 225 proved and credits this paper with the solution for general complex coefficients, independently of Kristiansen's proof, which has its own page, Kristiansen 1974. The page is dated by the first day of the issue month, since the paper's first posting carries no finer date.