Exact limsup of Faber polynomial norms when the maximal corner parameter Λk is not attained by any λj

Determine the exact value of L = limsup_{n→∞} ||F_n||_Γ for a piecewise Dini-smooth Jordan curve Γ with corners z_1,…,z_l having exterior angles λ_k π (0 ≤ λ_k ≤ 2), where Λ_k := max{λ_k, 2 − λ_k} and F_n denotes the nth Faber polynomial associated with Γ via the exterior conformal map. Specifically, resolve the case when no corner satisfies Λ_k = λ_j (i.e., when the condition max_{1≤k≤l} Λ_k = λ_j fails), for which Theorem 1.2 gives only the upper bound L ≤ max_{1≤k≤l} Λ_k but does not determine L exactly.

Background

The paper studies the asymptotic norms of Chebyshev and Faber polynomials on piecewise Dini-smooth Jordan curves with corner singularities, including cusps. For such a curve Γ, with exterior conformal map φ and Faber polynomials F_n (the polynomial part of φ(z)^n), the authors prove the bound limsup_{n→∞} ||F_n||_Γ ≤ max_k Λ_k, where each corner with exterior angle λ_k π contributes Λ_k = max{λ_k, 2 − λ_k}.

Moreover, they show that pointwise on Γ, φ(z)^{-n} F_n(z) → 1 away from the corners and equals λk at corner z_k. This implies equality in the global upper bound when there exists a corner z_j whose exterior angle parameter λ_j attains the maximum Λ_k, in which case limsup{n→∞} ||F_n||Γ = max_k Λ_k > 1. However, when the maximum is attained only by 2 − λ_k (e.g., all corners have exterior angle less than π), the theorem provides only an upper bound, and the exact value of the limsup remains undetermined. The circular lune C{1/2} is highlighted as a motivating example suggesting the limsup lies strictly between 1 and 3/2.