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Statement

Notation (pp. 125, 139): $E$ is the set where $|f|<1$; $\bar D_r$ is the closed disk $|z|\le r$; $S(r,n)$ is the maximum of the sum of the diameters of the components of $E$ over monic $f$ of degree $n$ with zeros in $\bar D_r$, and $S(r)$ the supremum of $S(r,n)$ over $n=1,2,\ldots$.

Theorem 9 (p. 140). "The function $S(r)$ has the following properties: if $0\leq r\leq1/2$, then $S(r)=2\sqrt{1+r^2}$; if $\varepsilon>0$ and $b$ is sufficiently small, then $S(1-b)>(1/2-\varepsilon)(1-e^{-1})\log1/b$."

So $S(r)$ is finite and explicit for small $r$ and tends to infinity as $r$ increases to $1$. The paper introduces $S(r)$ (p. 139) because $S(r,n)$, for $n\ge3$, "appears to be a discontinuous function of $r$, at $r=1$"; for unrestricted zeros it conjectures there that the sum of the diameters of the components of $E$ never exceeds $n2^{1/n}$, the value for $z^n-1$.

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Theorem 9 on p. 140, its proof on pp. 140--141. The copy read is identified on the source card.

Read depth. Claims checked: the statement and the definitions of $S(r,n)$ and $S(r)$ were read on the page images of pp. 139--140 on 2026-10-08; the proof was read for structure, not checked. Nothing here is independently reviewed.

Proof pointer

Pages 140--141. For $r\le1/2$, $E$ contains the disk of radius $1/2$ and is star-shaped, so the sum is the diameter of $E$; projecting the zeros onto a line through a longest chord of $\bar E$ can only enlarge the diameter, which reduces the first property to Theorem 2. For the lower bound, $Q(w)=(w^2-s^2)^q(w+s)$ with $s=\exp(-c_3/q^2)$ has a component of $E(Q)$ containing $[s/q,s]$ and avoiding the line $\operatorname{Re}w=s/2q$; then $f(z)=Q(z^n)$ has its zeros on $|z|=s^{1/n}$ and $n$ separate components each containing a segment longer than $r(1-q^{-1/n})$. Taking $n=[\log q]$ and $1-b=\exp(-c_3/(nq^2))$ gives the bound along a sequence $b_q$ with $b_{q+1}/b_q\to1$.

Dependencies

Theorem 2.

Bears on

No problem in the catalog is recorded here as concerning this theorem. It sums component diameters, while Problem 7 (#1048) asks for one large component.