In this paper we consider the
location of the zeros of a complex polynomial f(z) expressed as f(z) =∑k=0nakpk(z)
where {pk(z)} is a given sequence of polynomials of degree k whose zeros lie in a
prescribed region E. The principal theorem states that the zeros of f(z) are in the
interior of a Jordan curve S = {z;|F(z)| =Max(1,R)} where F maps the
complement of E onto |z| > 1 and R is the positive root of the equation
∑k=0n−1λk|ak|tk− λn|an|tn= 0, with λk> 0 depending on E only. Several
applications of this theorem are given. For example; if {pk(z)} is a sequence of
orthogonal polynomials on a ≦ z ≦ b, then we give an ellipse containing all the zeros
of ∑k=0nakpk(z).
