A note on some properties of the Perron root of nonnegative irreducible matrices

This paper deals with the Perron root of nonnegative irreducible matrices, all of whose entries are continuous bijective functions of some parameter vector. It is known that if the functions are log-convex, then the Perron root is a convex function of the parameter vector. In this paper, we strengthen this result by showing that the log-convexity property is also necessary when the Perron root is required to be convex for any nonnegative irreducible matrix. Furthermore, we show that a less restrictive requirement is sufficient when the matrix is confined to belong to two subsets of irreducible matrices. In particular, in case of positive semidefinite matrices, convexity is sufficient for the Perron root to be convex.