  We study the approximation of conformal mappings with the polynomials defined
by Keldysh and Lavrentiev from an extremal problem considered by Julia. These
polynomials converge uniformly on the closure of any Smirnov domain to the
conformal mapping of this domain onto a disk. We prove estimates for the rate
of such convergence on domains with piecewise analytic boundaries, expressed
through the smallest exterior angles at the boundary.
