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Statement

Notation (p. 125): $f$ is a monic polynomial (1) of degree $n$, $D$ the open unit disk, $\bar D$ its closure and $C$ the unit circle.

Theorem 12 (p. 145). "For any positive constant $c$, let $\mathfrak P(c)$ denote the class of polynomials (1) whose zeros lie in $\bar D$ and whose maximum modulus on $C$ is greater than $(1+c)^n$. Then there exist two constants $c_1=c_1(c)<1$ and $c_2=c_2(c)>0$ such that, for each $f$ in $\mathfrak P(c)$, the inequality $|f(z)|<c_1^n$ holds on a subset of $D$ whose measure is at least $c_2$."

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Theorem 12 on p. 145, its proof on pp. 145--147. The copy read is identified on the source card.

Read depth. Claims checked: the statement was read on the page image of p. 145 on 2026-10-08; the proof was read for structure, not checked. Nothing here is independently reviewed.

Proof pointer

Pages 145--147, by contradiction from a sequence $f_j$ with $|f_j(1)|>(1+c)^{n_j}$ for which the sets where $|f_j|<c_1^{n_j}$ have measure tending to $0$. Few zeros can lie in $|z|\le1-\varepsilon$, or $|f_j|$ would be exponentially small near the origin. The factor $g_j$ built from the remaining zeros is at least $(1+c/4)^{m_j}$ on a small disk $R$ near $1$ (inequality (7)). For $N_j$ rotated points $ze^{2\pi ip/N_j}$ the paper shows that the product of $|g_j|$ over them is below $2$ (inequality (8)), while the factors from points in $R$ are large; so a fixed proportion of the points must have $|g_j|<(1-c/8)^{c_5m_j}$, on every circle of a thin annulus.

Dependencies

None within the paper.

Bears on

No problem in the catalog is recorded here as concerning this theorem.