On the Crouzeix ratio for $N\times N$ matrices

The Crouzeix ratio $\psi (A)$ of an $N\times N$ complex matrix $A$ is the supremum of $\mathrm{\parallel }p(A)\mathrm{\parallel }$ taken over all polynomials $p$ such that $\mathrm{\mid }p\mathrm{\mid }\le 1$ on the numerical range of $A$. It is known that $\psi (A)\le 1+\sqrt{2}$, and it is conjectured that $\psi (A)\le 2$. In this note, we show that $\psi (A)\le C_N$, where $C_N$ is a constant depending only on $N$ and satisfying $C_N<1+\sqrt{2}$. The proof is based on a study of the continuity properties of the map $A\mapsto \psi (A)$.