On the convergence of best uniform deviations

If a function f is continuous on a closed Jordan curve $\Gamma$ and meromorphic inside $\Gamma$, then the polynomials of best uniform approximation to f on $\Gamma$ converge interior to $\Gamma$. Furthermore, the limit function can in each case be explicitly determined in terms of the mapping function for the interior of $\Gamma$. Applications and generalizations of this result are also given.