OMC for scalar multiples of unitaries

We establish the following case of the Determinantal Conjecture of Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973) 1137–1149] and de Oliveira [G.N. de Oliveira, Research problem: Normal matrices, Linear and Multilinear Algebra 12 (1982) 153–154]. Let A and B be unitary n × n matrices with prescribed eigenvalues a1, … , an and b1, … , bn, respectively. Then for any scalars t and sdet(tA-sB)∈co∏j=1n(taj-sbσ(j));σ∈Sn,where SnSn denotes the group of all permutations of {1, … , n} and co the convex hull taken in the complex plane.