Let G be the interior domain of a piecewise analytic Jordan curve without
cusps. Let {pn}n=0∞ be the sequence of polynomials that are
orthonormal over G with respect to the area measure, with each pn having
leading coefficient λn>0. N. Stylianopoulos has recently proven that
the asymptotic behavior of λn as n→∞ is given by πn+1λn2γ2n+2=1−αn, where
αn=O(1/n) as n→∞ and γ is the reciprocal of the
logarithmic capacity of the boundary ∂G. In this paper, we prove that
the O(1/n) estimate for the error term αn is, in general, best
possible, by exhibiting an example for which n→∞liminfnαn>0. The proof makes use of the Faber
polynomials, about which a conjecture is formulated.Comment: 7 page