Estimates for the asymptotic convergence factor of two intervals

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 $E$.