Approximation of $\operatorname{sgn}(x)$ on two symmetric intervals by rational functions

Solutions for the best approximation problems of $\operatorname{sgn}(x)$ over two symmetric intervals by odd rational functions with fixed poles are given in terms of conformal mappings onto special comb domains. Here the existence of the related conformal mapping is proved by using convexity of the comb domains along the imaginary axis (joint with D. V. Prokhorov)