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Setting

A rectifiable curve $\sigma$, $z=Z_\sigma(s)$ with $s\in[0,|\sigma|]$ the arc-length parameter, and its secant variation $\Psi(\sigma)$ are as on the Lemma 1 page (definitions (4)--(5), p. 52). For a curve with bounded rotation of the tangent (a Radon curve), $\Phi(\sigma)$ is the total variation of a single-valued branch of $\operatorname{Arg}Z_\sigma'(s)$, and $\Psi(\sigma)\le2\Psi_0(\sigma)\le2(\pi+\Phi(\sigma))$ (p. 53).

Statement

Theorem 2 (p. 56). Let $\sigma$ be rectifiable with $\Psi(\sigma)<\infty$, let $E$ be a compact subset of $\mathbb C$, and let $\mathcal E=\{s\in[0,|\sigma|]:Z_\sigma(s)\in E\}$. Then for every rational function $R$ of degree at most $n$ with no poles on $\sigma$,

$$\|R'\|_{L_1(E\cap\sigma)}:=\int_{\mathcal E}|R'(Z_\sigma(s))|\,ds \le n\Psi(\sigma)\|R\|_{C(E\cap\sigma)}. \tag{13}$$

Consequence (p. 57). For $\sigma\subset E$,

$$\operatorname{var}_\sigma R:=\int_0^{|\sigma|}|R'(Z_\sigma(s))|\,ds \le n\Psi(\sigma)\|R\|_{C(\sigma)} \le 2n(\pi+\Phi(\sigma))\|R\|_{C(\sigma)}, \tag{14}$$

the last inequality being meaningful for Radon curves. The first inequality in (14) is sharp: for $R(z)=z^n$ and the circle $\sigma=\{z=re^{it}:t\in[0,2\pi]\}$, $r>0$, one has $\Psi(\sigma)=2\pi$ and equality. For a circle $\sigma$, (13) reads $\|R'\|_{L_1(E\cap\sigma)}\le2\pi n\|R\|_{C(E\cap\sigma)}$, which the paper attributes to Dolzhenko (Anal. Math. 4 (1978)).

Proof pointer

P. 57. Apply Lemma 1a (p. 56; see the Lemma 1 page) with $E$ replaced by $R(E)$ and $\sigma$ by the image curve $R(\sigma)$; the image of the part of $\sigma$ over $\mathcal E$ has length, counted with multiplicity, $\|R'\|_{L_1(E\cap\sigma)}$. Lemma 4 (p. 56) gives $\Psi(R(\sigma))\le n\Psi(\sigma)$, since the argument of $(R-A)/(R-B)$ splits into the arguments of $n$ factors $(\zeta-a_j)/(\zeta-b_j)$ over the $A$- and $B$-points of $R$; and $\gamma(R(E\cap\sigma))\le\|R\|_{C(E\cap\sigma)}$, since a closed disc of radius $\rho$ has analytic capacity $\rho$.

Dependencies

Lemma 1a (p. 56), which rests on Lemma 1 and Remark 1 (pp. 53--56); Lemma 4 (p. 56).

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, (14), the sharpness example and the circle case read on the print; the proof (p. 57) and Lemmas 1a and 4 (p. 56) read but not checked step by step. Nothing here is independently reviewed.

Bears on

No Erdős problem directly. The theorem is the paper's companion estimate for rational functions; it does not bear on the length question of #114 beyond sharing the method of Theorem 1.