  The present paper is about Bernstein-type estimates for Jacobi polynomials
and their applications to various branches in mathematics. This is an old topic
but we want to add a new wrinkle by establishing some intriguing connections
with dispersive estimates for a certain class of Schr\"odinger equations whose
Hamiltonian is given by the generalized Laguerre operator. More precisely, we
show that dispersive estimates for the Schr\"odinger equation associated with
the generalized Laguerre operator are connected with Bernstein-type
inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi
polynomials to establish some new dispersive estimates. In turn, the optimal
dispersive decay estimates lead to new Bernstein-type inequalities.
