On the location of the zeros of certain composite polynomials (Q792478)

Let \(P(z)=\sum^{n}_{j=0}C(n,j)A_ jz^ j\) and \(Q(z)= \sum^{m}_{j=0} C(m,j)B_ jz^ j\) be two polynomials of degree \(n\) and \(m\) respectively, \(m\leq n\). In this paper we shall study the relative locationof the zeros of \(P(z)\) and \(Q(z)\) when the coefficients of these polynomials satisfyan apolar type relation and obtain some results. As an application of these results, we shall present certain generalizations of some results of Walsh, Szegö, de Bruijn and Kakeya.