---

Setting

Definitions, pp. 52--53. A rectifiable curve $\sigma$ is given by $z=Z_\sigma(s)$ with $s\in[0,|\sigma|]$ the arc-length parameter and $|\sigma|>0$; a point $z\in\sigma$ has multiplicity $k_\sigma(z)$, the number of $s$ with $Z_\sigma(s)=z$. A multiple union $L=[\sigma_1,\ldots,\sigma_m]$ of rectifiable curves, which may intersect, has length $|L|=|\sigma_1|+\cdots+|\sigma_m|$, counted with multiplicity.

For $z_1,z_2\notin\sigma$,

$$\Psi(\sigma;z_1,z_2)=\int_0^{|\sigma|}\Bigl|\,d\operatorname{Arg} \frac{Z_\sigma(s)-z_1}{Z_\sigma(s)-z_2}\Bigr|, \tag{4}$$

the variation along $\sigma$ of a continuous branch of the argument, where a factor $\zeta-z_j$ is replaced by $1$ when $z_j=\infty$. Then $\Psi(\sigma)=\sup\{\Psi(\sigma;z_1,z_2):z_1,z_2\notin\sigma\}$ and $\Psi_0(\sigma)=\sup\{\Psi(\sigma;z,\infty):z\notin\sigma\}$ (5), with $\Psi_0\le\Psi\le2\Psi_0$, $\Psi(\sigma)\ge2\pi$ always, and equality for circles, lines and non-degenerate segments (p. 52). For a multiple union, $\Psi(L;z_1,z_2)$ is the sum of the $\Psi(\sigma_k;z_1,z_2)$ and $\Psi(L)$ its supremum as in (5) (p. 53).

The analytic capacity of a compact $K\subset\mathbb C$ (p. 53) is the supremum of $|c_{-1}|$ over functions $f(z)=c_{-1}/z+c_{-2}/z^2+\cdots$ holomorphic on the component $G_\infty(K)$ of $\overline{\mathbb C}\setminus K$ containing $\infty$ with $\sup_{G_\infty(K)}|f|\le1$; and $\gamma(L)=\gamma(\bigcup_k\sigma_k)$, multiplicity not counted.

Statement

Lemma 1 (p. 53). If $L=[\sigma_1,\ldots,\sigma_m]$ is a multiple union of finitely many piecewise smooth curves and $\Psi(L)<\infty$, then

$$|L|=|\sigma_1|+\cdots+|\sigma_m|\le\Psi(L)\,\gamma(L). \tag{6}$$

For every circle $L=\sigma$, (6) is an equality (p. 53). The paper says (p. 53) that Lemmas 1 and 4 appeared, in a somewhat different form and without proof, in the author's 1985 note in Dokl. Akad. Nauk SSSR.

Extensions (pp. 55--56). Remark 1 states that Lemma 1 holds for arbitrary rectifiable $L$, by inscribing polygonal lines and using the upper semicontinuity of analytic capacity. Lemma 1a (p. 56), which Remark 2 derives from Lemma 1 and whose proof applies Lemma 1 through Remark 1: if $\sigma$ is rectifiable with $\Psi(\sigma)<\infty$, $E$ is a compact subset of $\mathbb C$ and $\mathcal E=\{s\in[0,|\sigma|]:Z_\sigma(s)\in E\}$, then $|E\cap\sigma|:=\operatorname{mes}_1\mathcal E\le\Psi(\sigma)\gamma(E\cap\sigma)$ (11), the length counted with the multiplicity $k_\sigma(z)$.

Proof pointer

P. 54. Orient each $\sigma_k$ so that $\operatorname{Re}d\zeta\ge0$; the multiple projection of $L$ to the real axis then has length at most $|\int_{L^+}d\zeta|$ (7). The function $w_0(z)=\int_{L^+}d\zeta/(\zeta-z)$ maps $G_\infty(L)$ into a horizontal strip of width at most $\Psi(L)$; scaling by $\pi/\Psi(L)$, exponentiating and a Möbius map (8) send $G_\infty(L)$ into the unit disc with $\infty\mapsto0$, and the definition of analytic capacity bounds $|\int_{L^+}d\zeta|$ by $2\Psi(L)\gamma(L)/\pi$. The same bound holds for the projection to every line, and Cauchy's formula $|L|=\frac12\int_0^\pi|\Pi_\varphi|\,d\varphi$ gives (6).

Dependencies

The definitions (4)--(5) and of analytic capacity (pp. 52--53); Cauchy's projection formula, cited from Dolzhenko.

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 definitions it uses, Remark 1 and Lemma 1a read on the print; the proof (p. 54) read for its structure, not checked step by step. Nothing here is independently reviewed.

Bears on

• #114: background only. The lemma is the step of Theorem 1 that turns bounds on $\Psi$ and $\gamma$ of a lemniscate into the length bound $2\pi n$ at $r=1$; on its own it says nothing about which polynomial maximises the length.