Let A be an n × n matrix, q(A) = min{|λ|: λ ∈ σ(A)} and σ(A) denote the spectrum of A. From Fiedler and Markham [Linear Algebra Appl. 101 (1988) 1], Song [Linear Algebra Appl. 305 (2000) 99] and Yong [Linear Algebra Appl. 320 (2000) 167], for the Hadamard products of n×n M-matrices and their inverses, the infimum of q(A◦ A−1) is 2/n. In this paper the following results are presented: if q(Ak◦ A−1 k ) tends to the infimum 2/n for n×n (n>2) M-matrices Ak, k=1, 2,..., then the spectral radius ρ(Jk) of the Jacobi iterative matrix of Ak tends to 1. That is, if q(A◦ A−1) is close to 2/n, then ρ(J) is close to 1; and another lower bound is given for A being an n×n M-matrix, q(A◦ A−1) max1−ρ(J)2, +ρ(J)1 n+21+(n−1)ρ(J) 1n+2 where ρ(J) is the spectral radius of the Jacobi iterative matrix of A. Furthermore, if A is an H-matrix, then q(A◦A−1) (1−ρ(Jm(A))2)/(1+ρ(Jm(A))2), where ρ(Jm(A)) is the spectral radius of the Jacobi iterative matrix of the comparison matrix m(A).