Approximation to the Function $z^{\alpha}$ by Rational Fractions in a Domain with Zero External Angle

Rational approximations to the function $z^{\alpha}$, $\alpha\in\mathbb{R}\setminus\mathbb{Z}$, were studied by Newman, Gonchar, Bulanov, Vyacheslavov, Andersson, Stahl, and others. The present paper deals with the order of best rational approximations to this function in a domain with zero external angle and vertex at the point $z=0$. In particular, the obtained results show that the conditions imposed on the boundary of the domain in the Jackson-type inequality proved by the author in 2001 for the best rational approximations in Smirnov spaces cannot be weakened significantly.