An upper bound on the characteristic polynomial of a nonnegative matrix leading to a proof of the Boyle-Handelman conjecture.

In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if $A$ is an $(n+1)times (n+1)$ nonnegative matrix whose {bf nonzero} eigenvalues are: $lambda _0 geq |lambda _i|$, $i=1,ldots ,r$, $r leq n$, then for all $x geq lambda _0$, begin {equation} prod _{i=0}^{r} (x-lambda _i) leq x^{r+1}-lambda _0^{r+1}.tag *{$(ast )$} endequation }}}par To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when $2(r+1)geq (n+1)$, while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when $nleq 4$ and when the spectrum of $A$ is real. They also showed that the conjecture is asymptotically true with the dimension. par Here we prove a slightly stronger inequality than in $(ast )$, from which it follows that the Boyle--Handelman conjecture is true. Actually, we do not start from the assumption that the $lambda _i$'s are eigenvalues of a nonnegative matrix, but that $lambda _1,ldots , lambda _{r+1}$ satisfy $lambda _0geq |lambda _i|$, $i=1,ldots , r$, and the trace conditions: begin {equation} sum _{i=0}^{r} lambda _i^k geq 0, mbox {for all} k geq 1.tag *{$(ast ast )$} endequation }}} A strong form of the Boyle--Handelman conjecture, conjectured in 2002 by the present authors, says that ($*$) continues to hold if the trace inequalities in ($**$) hold only for $k=1,ldots ,r$. We further improve here on earlier results of the authors concerning this stronger form of the Boyle--Handelman conjecture. [ABSTRACT FROM AUTHOR]