Proof of a conjecture on numerical ranges of weighted cyclic matrices

Recall that the n-by-n weighted cyclic matrix with weights a₁,…,a_n(∈C) is the matrix C(a₁,…,a_n)=[0a₁0⋱⋱a_n−1a_n0], and W(C(a₁,…,a_n)) is the numerical range of C(a₁,…,a_n). Let S_n be the symmetric group on {1,…,n}. In [2], Chien et al. conjecture that if |a₁|≥|a₂|≥…≥|a_n| then W(C(a_η(1),…,a_η(n)))⊆W(C(a_σn(1),…,a_σn(n))) for any permutation η∈S_n, where σ_n∈S_n is defined by (σ_n(1),…,σ_n(n))={(n−1,…,4,2,1,3,5,…,n−2,n)if n is odd,(n−2,…,4,2,1,3,5,…,n−1,n)if n is even. In this note, we settle the conjecture in the affirmative.