  In this paper, we uncover an intriguing algebra property of an element
symmetric polynomial. By this property, we establish the longtime existence and
convergence of a locally constrained flow, thereby some families of geometric
inequalities in sphere can be derived. Meanwhile, a new family of ``three
terms'' geometric inequalities involving two weighted curvature integrals and
one quermassintegral are proved. Unlike hyperbolic spaces, we also obtain an
inverse weighted geometric inequality in sphere.
