I know I can bound the spectral radius of a matrix by a norm of the matrix. Are there other approaches? In my case, I have a positive square matrix J(v) that depends on some fixed parameters and the vector v that lives in the simplex (i.e., $v_i\geq0,\forall i$ and $\sum_i v_i =1$), and I need to show that $$\max_v \rho(J(v)) \leq 1.$$
Simulations reveal that this is true, but also that well known norms (i.e., infinity norm or spectral norm) don't give me the right bounds. Is proof by contradiction useful? I am going to leave this question vague because the details are somewhat complicated and probably not of much interest here, but suggestions and examples would be most welcomed.