Let P(z) = Σj=0nC(n,j)Ajzj
and Q(z) = Σj=0nC(n,j)Bjzj, AnBn≠0, be two polynomials of the same degree n. If
P(z) and Q(z) are apolar and if one of them has all its zeros in a circular region C,
then according to a famous result known as Grace’s theorem, the other will have at
least one zero in C. In this paper we propose to relax the condition that P(z) and
Q(z) are of the same degree. Instead, we will assume P(z) and Q(z) to be the
polynomials of arbitrary degree n and m respectively, m ≦ n, with their coefficients
satisfying an apolar type relation and obtain certain generalizations of Grace’s
theorem for the case when the circular region C is a circle |z| = r. As an
application of these results, we also generalize some results of Szegő, Cohn and
Egerváry.
