  We derive upper bounds for the potential energy of spherical designs of
cardinality close to the Delsarte-Goethals-Seidel bound. These bounds are
obtained by linear programming with the use of the Hermite interpolating
polynomial of the potential function in suitable nodes. Numerical computations
show that the results are quite close to certain lower energy bounds confirming
that spherical designs are, in a sense, energy efficient. \end{abstract}
