Abstarct: In this thesis,we present certain interesting refinements of a well-known EnestromKakeya theorem in the theory of distribution of zeros of polynomials which among other things also improve upon some results of Asis and Mohammad, Govil and Rehman and others. We obtain the region containing all the zeross of aclass of analytic functions whose coefficients are subject to certain conditions. We extend the Enestrom-Kakeya theorem and its refinement by Hurwitz to polynomial matrices H(z) with positive semidefinitc coefficients. We determine an annular region containing the zeros of det H(z).A stability result for systems of linear difference equation is given as an application.