  Recently we introduced a new Bernstein-type operator using P\'olya's urn
model with negative replacement, and we showed that it satisfies a
Popoviciu-type inequality with a constant slightly larger than that of the
corresponding inequality for the classical Bernstein operator.
  In the present paper we prove an inequality for the rising factorial (of
independent interest), and we use it in order to show that the constant in the
Popoviciu inequality for the new operator is in fact smaller than the
corresponding constant for the Bernstein operator.
