An improvement of an inequality of fiedler leading to a new conjecture on nonnegative matrices

Suppose that A is an n × n nonnegative matrix whose eigenvalues are λ = ρ(A), λ ₂,..., λ _n. Fiedler and others have shown that det(λI - A) ≤ λ ⁿ - ρ ⁿ, for all λ > ρ, with equality for any such λ if and only if A is the simple cycle matrix. Let a _i be the signed sum of the determinants of the principal submatrices of A of order i × i, i=1,..., n - 1. We use similar techniques to Fiedler to show that Fiedler's inequality can be strengthened to: det(λI - A) + ∑ _i=1 ^n-1 ρ ^n-2i |a _i| (λ - ρ) ⁱ ≤ λ ⁿ - ρ ⁿ, for all λ ≥ ρ. We use this inequality to derive the inequality that: ∏ ₂ ^k(ρ - λ _i) ≤ ρ ^n-2 ∑ _i=2 ⁿ (ρ - λ _i). In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of A: If λ ₁ = ρ(A), λ ₂,..., λ _k are (all) the nonzero eigenvalues of A, then ∏ ₂ ^k(ρ - λ _i) ≤ ρ ^k-2 ∑ _i=2 ^k (ρ - λ). We prove this conjecture for the case when the spectrum of A is real.