Spectral bounds for certain special type of rational matrices

We derive bounds on the moduli of the eigenvalues of special type of rational matrices using the following techniques/methods: (1) the Bauer-Fike theorem on an associated block matrix of the given rational matrix, (2) by associating a real rational function, along with Rouch$\text{\'e}$ theorem for the rational matrix and (3) by 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.