On the leading coefficient of polynomials orthogonal over domains with corners

Let $G$ be the interior domain of a piecewise analytic Jordan curve without cusps. Let $\{p_n{\}}_{n=0}^{\mathrm{\infty }}$ be the sequence of polynomials that are orthonormal over $G$ with respect to the area measure, with each $p_n$ having leading coefficient ${\lambda }_n>0$. N. Stylianopoulos has recently proven that the asymptotic behavior of ${\lambda }_n$ as $n\to \mathrm{\infty }$ is given by $$\frac{n+1}{\pi }\frac{{\gamma }^{2n+2}}{{\lambda }_n^2}=1-{\alpha }_n,$$ where ${\alpha }_n=O(1\mathrm{/}n)$ as $n\to \mathrm{\infty }$ and $\gamma$ is the reciprocal of the logarithmic capacity of the boundary $\mathrm{\partial }G$. In this paper, we prove that the $O(1\mathrm{/}n)$ estimate for the error term ${\alpha }_n$ is, in general, best possible, by exhibiting an example for which $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}n{\alpha }_n>0.$$ The proof makes use of the Faber polynomials, about which a conjecture is formulated.