  Motivated by the problem of optimal portfolio liquidation under transient
price impact, we study the minimization of energy functionals with completely
monotone displacement kernel under an integral constraint. The corresponding
minimizers can be characterized by Fredholm integral equations of the second
type with constant free term. Our main result states that minimizers are
analytic and have a power series development in terms of even powers of the
distance to the midpoint of the domain of definition and with nonnegative
coefficients. We show moreover that our minimization problem is equivalent to
the minimization of the energy functional under a nonnegativity constraint.
