  We present a Suffridge-like extension of the Grace-Szeg\"o convolution
theorem for polynomials and entire functions with only real zeros. Our results
can also be seen as a $q$-extension of P\'olya's and Schur's characterization
of multiplier sequences. As a limit case we obtain a new characterization of
all log-concave sequences in terms of the zero location of certain associated
polynomials. Our results also lead to an extension of Ruscheweyh's convolution
lemma for functions which are analytic in the unit disk and to new necessary
conditions for the validity of the Riemann Conjecture.
