  In this paper, we establish several results related to Crouzeix's conjecture.
We show that the conjecture holds for contractions with eigenvalues that are
sufficiently well-separated. This separation is measured by the so-called
separation constant, which is defined in terms of the pseudohyperbolic metric.
Moreover, we study general properties of related extremal functions and
associated vectors. Throughout, compressions of the shift serve as illustrating
examples which also allow for refined results.
