Crouzeix's conjecture holds for tridiagonal 3x3 matrices with elliptic numerical range centered at an eigenvalue

M. Crouzeix formulated the following conjecture in (Integral EquationsOperator Theory 48, 2004, 461--477): For every square matrix A and everypolynomial p,

∥p(A)∥≤2maxz∈W(A)|p(z)|, where W(A) is the numerical rangeof A. We show that the conjecture holds in its strong, completely boundedform, i.e., where p above is allowed to be any matrix-valued polynomial, forall tridiagonal 3×3 matrices with constant main diagonal: ⎡⎣⎢ac10b1ac20b2a⎤⎦⎥,a,bk,ck∈C, or equivalently, for all complex 3×3 matriceswith elliptic numerical range and one eigenvalue at the center of the ellipse.We also extend the main result of D. Choi in (Linear Algebra Appl. 438,3247--3257) slightly.