Two inequalities for the Perron root (Q1089066)

If A and B are irreducible, nonnegative \(n\times n\) matrices with a common right and a common left eigenvector corresponding to their respective spectral radii r(A), r(B), then it is shown that \(r(tA+(1- t)B^+)\geq tr(A)+(1-t)r(B)\), where \(0\leq t\leq 1\) and \(B^+\) denotes the transpose of B. Previous inequalities due to \textit{B. W. Levinger} [Notices Am. Math. Soc. 17, 260 (1970)] and \textit{S. Friedland} and \textit{S. Karlin} [Duke II Math. J. 42, 459-490 (1975; Zbl 0373.15008)] are generalized.