  We prove a multiplier version of the Bernstein inequality on the complex
sphere. Included in this is a new result relating a bivariate sum involving
Jacobi polynomials and Gegenbauer polynomials, which relates the sum of
reproducing kernels on spaces of polynomials irreducibly invariant under the
unitary group, with the reproducing kernel of the sum of these spaces, which is
irreducibly invariant under the action of the orthogonal group.
