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Statement

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

Theorem 8 (p. 139). "Let $f(z)=\prod_1^n(z-x_\nu)$, where the $x_\nu$ are real and not all equal; and suppose that $a$ is a positive number such that $|f(a)|=|f(-a)|$. Then $|f(ae^{i\vartheta})|>|f(a)|$, except when $\sin\vartheta=0$."

Corollary (p. 140). "If all the $z_\nu$ are real, then the sum of the diameters of the components of $E$ is $|E\cap L|$."

The paper calls Theorem 8 a preliminary proposition for Section 6 (p. 139).

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Theorem 8 on p. 139, the Corollary and the proof on p. 140. The copy read is identified on the source card.

Read depth. Claims checked: the theorem and corollary were read on the page images of pp. 139--140 on 2026-10-08; the proof was read and its steps followed, not independently checked. The paper gives no separate proof of the corollary. Nothing here is independently reviewed.

Proof pointer

Page 140. It suffices to take $0<\vartheta<\pi$. With $H(\vartheta)=2\log|f(ae^{i\vartheta})|=\sum\log(a^2+x_\nu^2-2ax_\nu\cos\vartheta)$, one has $H'(\vartheta)=2a\sin\vartheta\sum x_\nu/|ae^{i\vartheta}-x_\nu|^2$, and each term $x_\nu/|ae^{i\vartheta}-x_\nu|^2$ decreases in $\vartheta$ whatever the sign of $x_\nu$. So $H$ has no interior local minimum on $(0,\pi)$, and its values there exceed the common endpoint value $H(0)=H(\pi)$.

Dependencies

None.

Bears on

• #1040: the paper says (p. 136) that the line-segment case of Problem 4 follows from Chebyshev ("Tchebycheff") polynomials together with Theorem 8; it gives no further detail.