
## 3. Barycentric approximation to solve the conjecture

1.     The motivation to use barycentric approximation comes from Pearcy’s proof showing that $W(A)\subset \mathbb{D}$ implies that $W(A^{k})\subset \mathbb{D}$ for any $k\in\mathbb{N}$ (1965).

#### Problem 3.1.

[N. Trefethen] Try barycentric approximation for more general $W(A)$ in order to show that $W(f(A))\subset \frac{5}{4}\mathbb{D}$ for $\|f\|_{\infty,W(A)}=1$.
1. Remark. [T. Ransford] T. Ransford has done this in the case of $W(A)\subset \mathbb{D}$ and Blaschke products $f$.

Cite this as: AimPL: Crouzeix's conjecture, available at http://aimpl.org/crouzeix.