Suffridge's Convolution Theorem for Polynomials and Entire Functions Having Only Real Zeros

Abstract

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.