  Let $E$ be the union of two real intervals not containing zero. Then
$L_n^r(E)$ denotes the supremum norm of that polynomial $P_n$ of degree less
than or equal to $n$, which is minimal with respect to the supremum norm
provided that $P_n(0)=1$. It is well known that the limit
$\kappa(E):=\lim_{n\to\infty}\sqrt[n]{L_n^r(E)}$ exists, where $\kappa(E)$ is
called the asymptotic convergence factor, since it plays a crucial role for
certain iterative methods solving large-scale matrix problems. The factor
$\kappa(E)$ can be expressed with the help of Jacobi's elliptic and theta
functions, where this representation is very involved. In this paper, we give
precise upper and lower bounds for $\kappa(E)$ in terms of elementary functions
of the endpoints of $E$.
