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Statement

Notation (p. 125): $E$ is the set where $|f|<1$.

Theorem 11 (p. 143). "If the $z_\nu$ lie in a disk of radius

$$r_0=\frac{\sin\pi/8}{1+\sin\pi/8},$$

then the set $E$ is convex."

Numerically $r_0\approx0.2768$. The paper adds (p. 145) that the example $f(z)=(z-r)^m(z+r)$ with $m$ large shows that the theorem fails if $r_0$ is replaced by a constant greater than $1/2$; it does not settle the constants between $r_0$ and $1/2$. The theorem places no condition on the center of the disk; the proof takes it at the origin (p. 143).

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Theorem 11 on p. 143, its proof on pp. 143--145 and the example on p. 145. The copy read is identified on the source card.

Read depth. Claims checked: the statement and the example were read on the page images of pp. 143 and 145 on 2026-10-08; the proof was read for structure, not checked. Nothing here is independently reviewed.

Proof pointer

Pages 143--145. With the zeros in $\bar D_r$, the proof shows that the lemniscate $|f|=1$ has no point of inflection when $r\le r_0$. At a supposed inflection point $z$, expand $|f|$ along the curve to second order in the arc parameter $t$: the first-order term vanishes because $|f|$ is constant on the curve, which turns the second-order coefficient into $-\sum\cos2\alpha_\nu/(2\rho_\nu^2)$, with $\rho_\nu=|z-z_\nu|$ and $\alpha_\nu$ the angle between the line $z_\nu z$ and the tangent. Since $|z|\ge1-r$, all the $\alpha_\nu$ lie within $2\sin^{-1}(r/(1-r))$ of each other, and with the vanishing first-order term this puts every $2\alpha_\nu$ within $4\sin^{-1}(r/(1-r))$ of $\pi$. When $\sin^{-1}(r/(1-r))\le\pi/8$, that is $r\le r_0$, the coefficient is positive, contradicting $|f|=1$ along the curve.

Dependencies

None.

Bears on

• #1047: the paper calls Grunsky's question, Problem 16, "related to our theorem" (p. 145). The theorem concerns the level $1$ with all zeros in one small disk, the problem the loops around distinct zeros at a small level; the paper derives neither from the other.