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Setting

Notation (1), p. 51. For a natural number $n$, complex numbers $c_0,\ldots,c_{n-1}$ and $r>0$,

$$P_n(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_1z+c_0,\qquad L(P_n,r)=\{z:|P_n(z)|=r^n\}.$$

The paper notes (p. 51) that $L(P_n,r)$ consists of $m\le n$ closed Jordan curves $\sigma_1,\ldots,\sigma_m$ with pairwise disjoint interiors, which may meet only at zeros of $P_n'$, and its length is $|L(P_n,r)|=|\sigma_1|+\cdots+|\sigma_m|$.

Statement

Theorem 1 (p. 52). For every such $P_n$ and $r>0$,

$$|L(P_n,r)|\le 2\pi nr. \tag{3}$$

The paper places this against earlier upper bounds (p. 51): Dolzhenko's $|L(P_n,r)|\le 4\pi nr$, Pommerenke's $|L(P_n,1)|\le 74n^2$, Borwein's $|L(P_n,1)|\le 8\pi en$ and Eremenko and Hayman's $|L(P_n,1)|\le 9.173n$.

The conjectured extremal case (p. 52). The paper records as a natural conjecture, crediting it to its references [2]--[4] and [6], that among all $L(P_n,1)$ the Bernoulli-type lemniscate $L(B_n,1)$, $B_n(z)=z^n-1$, is longest, and computes in polar coordinates

$$|L(B_n,1)|=2^{1+1/n}\int_0^{\pi/2}\cos^{1/n-1}t\,dt =2^{1/n}B\Bigl(\frac12,\frac1{2n}\Bigr) =2^{1/n}\sqrt\pi\,\frac{\Gamma(\frac1{2n})}{\Gamma(\frac12+\frac1{2n})} =2n+4\ln2+O\Bigl(\frac1n\Bigr).$$

Proof pointer

P. 52: the inequality follows at once from Lemmas 1--3. By Lemma 1, $|L|\le\Psi(L)\gamma(L)$, the secant variation times the analytic capacity. Lemma 2 (p. 55) gives $\gamma(L(P_n,r))\le r$: with $f$ the Ahlfors function of the unbounded complementary component, $f^nP_n$ is bounded by $r^n$ there and equals $\gamma^n$ at infinity. Lemma 3 (p. 55) gives $\Psi(L(P_n,r))\le 2\pi n$ for every $r>0$: after a Möbius change of variable the image of the lemniscate is a level line of a rational function of degree at most $n$, which meets each line through the origin in at most $2n$ points. Multiplying the two bounds gives (3).

Dependencies

Lemma 1 (p. 53); Lemma 2 and Lemma 3 (p. 55); the definitions of $\Psi$ and $\gamma$ (pp. 52--53).

Source. V. I. Danchenko, The lengths of lemniscates. Variations of rational functions, Mat. Sb. 198 (2007), no. 8, 51--58 (in Russian); pages are the journal's, as on the source card.

Read depth. Claims checked: the statement, the setting (1) and the Bernoulli computation read on the print; the proofs of Lemmas 1--3 (pp. 53--55) read but not checked step by step. Nothing here is independently reviewed.

Bears on

• #114: at $r=1$ the theorem bounds the length of $\{z:|p(z)|=1\}$ by $2\pi n$ for every monic $p$ of degree $n$. The conjectured maximiser $z^n-1$ has length $2n+4\ln2+O(1/n)$, so the bound exceeds it by a factor tending to $\pi$ as $n$ grows; the theorem does not decide whether $z^n-1$ maximises the length.