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Statement

Remark after Lemma 5 (p. 5). From Lemma 5 the paper concludes that $z^2+1$ is extremal for $d=2$: no monic quadratic $p$ has $|E(p)|$ larger than $|E(z^2+1)|$. The level set $\{z:|z^2+1|=1\}$ is the Bernoulli lemniscate (also one of Cassini's ovals), and its length is the elliptic integral

$$2^{3/2}\int_{-1}^{1}\frac{dx}{\sqrt{1-x^4}}\approx7.416$$

(p. 5). The abstract (p. 1) states the result as: for $d=2$ the extremal level set is the Bernoulli lemniscate.

The remark gives no further argument. The deduction it leaves to the reader: a monic quadratic has one critical point, so the extremal polynomial of Lemma 5 is $(z-c)^2+a$ with $|a|=1$, which is $z^2+1$ up to translation and rotation, operations that keep the length.

Source. Alexandre Eremenko and Walter Hayman, On the length of lemniscates, Michigan Math. J. 46 (1999), no. 2, 409--415, DOI 10.1307/mmj/1030132418; page numbers are those of the authors' corrected preprint (pp. 1--9) named on the source card, not the journal's pagination.

Read depth. Claims checked: the remark was read on the print. Nothing here is independently reviewed.

Proof pointer

P. 5, by Lemma 5 as above.

Dependencies

Lemma 4 and Lemma 5.

Bears on

• #114: the case $n=2$. Since $z^2+1$ and $z^2-1$ differ by the rotation $z\mapsto iz$, it says that $z^2-1$ maximizes the length among monic quadratics; the claim page Eremenko–Hayman 1999 records this. It settles no other degree.