Spectral bounds for certain special type of rational matrices

We derive bounds on the moduli of eigenvalues of certain special type of rational matrices, using the following techniques/methods: (1) an upper bound is obtained using the Bauer-Fike theorem on an associated block matrix of the given rational matrix, (2) a lower bound is obtained by associating a real rational function, along with Rouch$\text{\'e}$'s theorem for the rational matrix and (3) an upper bound is also obtained using a numerical radius inequality for a block matrix for the rational matrix. These bounds are compared when the coefficients are unitary matrices. Numerical examples are given to illustrate the results obtained.