Asymptotic behavior of the least deviations of the function $\operatorname{sgn}x$ from rational functions

It is shown that the best uniform approximation of $\operatorname{sgn}x$ by rational functions of order at most $n$ on the union of the two intervals $[-1,-\delta]\cup[\delta,1]$ ($0<\delta<1$) does not exceed
$$ e^{\frac{\pi^2}2}\exp\biggl\{-\frac{\pi^2}2\frac n{\ln\frac1\delta+2\ln\ln\frac e\delta+2}\biggr\}. $$

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