  In 1976 Suffridge proved an intruiging theorem regarding the convolution of
polynomials with zeros only on the unit circle. His result generalizes a
special case of the fundamental Grace-Szeg\"o convolution theorem, but so far
it is an open problem whether there is a Suffridge-like extension of the
general Grace-Szeg\"o convolution theorem. In this paper we try to approach
this question from two different directions: First, we show that Suffridge's
convolution theorem holds for a certain class of polynomials with zeros in the
unit disk and thus obtain an extension of one further special case of the
Grace-Szeg\"o convolution theorem. Second, we present non-circular zero domains
which stay invariant under the Grace-Szeg\"o convolution hoping that this will
lead to further analogs of Suffridge's convolution theorem.
