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](http://www.turpion.org/php/full/infoFT.phtml?journal_id=sm&paper_id=1107))


$$Z\_{k}(E,F) \geq e^{-k/C(E,F)},\quad k\geq 0,$$


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](http://www.turpion.org/php/full/infoFT.phtml?journal_id=sm&paper_id=1107))
$$\lim\_{k\rightarrow\infty}Z\_k(E,F)^{1/k} = e^{-1/C(E,F)}.$$


* **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](https://arxiv.org/abs/1609.09494)


$$Z\_k([-b,-a],[a,b])\leq 4e^{-k/C([-b,-a],[a,b])}.$$


(I am asking for an upper bound for general disjoint plates (E,F).)



An upper bound on Zolotarev numbers
T. Ganelius has shown inequalities of the type
$$
Z\_k(E,F)\leq const. e^{-k/C(E,F)},\qquad k=1,2,\ldots,\qquad (\*)
$$
under additional assumptions on $E$ and $F$, see


[1] T. Ganelius, Some extremal functions and approximation. Fourier analysis and approximation theory (Proc. Colloq., Budapest, 1976), Vol. I, pp. 371–381, Colloq. Math. Soc. János Bolyai, 19, North-Holland, Amsterdam-New York, 1978. (MR0540314)


[2] T. Ganelius, Rational functions, capacities and approximation. Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979), pp. 409–414, Academic Press, London-New York, 1980. (MR0623483)


From the introduction of [2] : "it was shown in [1] that if E and F are finite unions of sets of bounded rotation, then there is a constant, depending only on the geometric configuration, such that (\*) holds true. The dependence on the configuration is cumbersome and it has been suggested by Gonchar that there should be better results at least for ring condensers. In this note (i.e. [2]) we give a result of this type."


