Let $k$ be an integer and disjoint closed sets $E,F\subset\mathbb{R}^2$. Consider the Zolotarev number
$$Z_k(E,F):= \inf_{r\in\mathcal{R}_{k,k}}\frac{\sup_{z\in E}|r(z)|}{\inf_{z\in F}|r(z)|},$$
where $\mathcal{R}_{k,k}$ is the space of degree $(k,k)$ rational functions.
In 1969, Gonchar showed that (paper here)
$$Z_{k}(E,F) \geq e^{-k/C(E,F)},\quad k\geq 0,$$
where $C(E,F)$ is the logarithmic capacity of the plates $(E,F)$.
My question is: What upper bounds on $Z_k(E,F)$ are known? I am hoping for a bound of the form
$$ Z_k(E,F) \leq {\rm const.} \text{ } e^{-k/C(E,F)}, \quad k\geq 0$$
for some explicitly known constant.
Here, are two closely related results that may be useful.
- The lower bound is asymptotically tight
For disjoint closed sets $E,F\subset\mathbb{R}^2$, we have (see here) $$\lim_{k\rightarrow\infty}Z_k(E,F)^{1/k} = e^{-1/C(E,F)}.$$
- Explicit case when $E$ is a real interval and $F = -E$
When $0<a<b<\infty$ and $E = [-b,-a]$ and $F=[a,b]$, we know from Zolotarev's work (see sec. 51 in "Elements of the Theory of Elliptic Functions" by Akhiezer) that
$$C([-b,-a],[a,b]) = \frac{\mu(a/b)}{\pi^2},$$
where $\mu(\lambda) = \tfrac{\pi}{2}K(\sqrt{1-\lambda^2})/K(\lambda)$ is the Grotsch ring function and $K(\cdot)$ is the complete elliptic integral of the first kind
$$K(\lambda) = \int_{0}^1 \frac{1}{\sqrt{(1-t^2)(1-\lambda^2t^2)}}dt, \qquad 0\leq \lambda\leq 1.$$ Moreover, the upper bound is see Cor. 3.2 here
$$Z_k([-b,-a],[a,b])\leq 4e^{-k/C([-b,-a],[a,b])}.$$
(I am asking for an upper bound for general disjoint plates (E,F).)
