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Statement

Notation (p. 125): $f$ is a monic polynomial (1), $E$ the set where $|f|<1$, $D$ the open unit disk, $\bar D$ its closure and $C$ the unit circle; $|\cdot|$ on plane sets is area.

Theorem 4 (p. 133). "If all $z_\nu$ lie in $\bar D$, then $|E|\leq4\{\pi|E\cap D|\}^{1/2}$."

Corollary (p. 133). "For the class of functions (1) with all $z_\nu$ on $C$, $\inf|E|=0$."

The corollary combines the theorem with what the paper draws from MacLane's theorem (its [5], Theorem A) on p. 133: for zeros on $C$, $\inf|E\cap D|=0$. The paper records (pp. 132--133) that this refutes the statement of Erdős's 1940 note (its [2], p. 958) that the area of $E$ exceeds a positive universal constant when the zeros lie in $\bar D$, a statement it says rested on an unpublished erroneous argument.

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Theorem 4, the Corollary and the proof on p. 133. The copy read is identified on the source card.

Read depth. Claims checked: the theorem, the corollary and the MacLane paragraph were read on the page images of pp. 132--133 on 2026-10-08; the proof was read and its steps followed, not independently checked. Nothing here is independently reviewed.

Proof pointer

Page 133. With zeros in $\bar D$, $E$ lies in $|z|\le2$, and $|f(re^{i\vartheta})|<|f((2-r)e^{i\vartheta})|$ for $0<r<1$, so every point of $E$ outside $D$ is the image of a point of $E\cap D$ under $w(re^{i\vartheta})=(2-r)e^{i\vartheta}$. That map multiplies area by $(2-r)/r$, a decreasing function of $r$, so a set of area $S$ in $D$ has an image of area at most $4\sqrt{\pi S}-S$, the value for the disk of area $S$ about the origin. Adding $|E\cap D|$ gives the theorem.

Dependencies

MacLane's Theorem A (the paper's [5]) for the corollary; nothing else in the paper.

Bears on

• #116: the least area $\alpha_n$ of Problem 2 is taken over zeros in $\bar D$, which includes zeros on $C$, so the corollary gives $\inf_n\alpha_n=0$; the problem asks how fast it can decrease. The step from the corollary to $\alpha_n$ is drawn here.
• #1040: for $F=C$ the corollary is $\mu(C)=0$, and since $C\subset\bar D$, $\mu(\bar D)=0$ follows; both sets have transfinite diameter $1$. The paper itself says (p. 136) that the disk case of Problem 4 follows from Theorem 4.