A multiplier version of the Bernstein inequality on the complex sphere

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 unitary group.