On Cauchy-type bounds for the eigenvalues of a special class of matrix polynomials

Let $\mathbb{C}^{m\times m}$ be the set of all $m\times m$ matrices whose entries are in $\mathbb{C},$ the set of complex numbers. Then $P(z):=\sum\limits_{j=0}^nA_jz^j,~A_j\in \mathbb{C}^{m\times m},~0\leq j\leq n$ is called a matrix polynomial. If $A_{n}\neq 0$, then $P(z)$ is said to be a matrix polynomial of degree $n.$ In this paper we prove some results for the bound estimates of the eigenvalues of some lacunary type of matrix polynomials.