  We prove a new inequality which improves on the classical Hardy inequality in
the sense that a nonlinear integral quantity with super-quadratic growth, which
is computed with respect to an inverse square weight, is controlled by the
energy. This inequality differs from standard logarithmic Sobolev inequalities
in the sense that the measure is neither Lebesgue's measure nor a probability
measure. All terms are scale invariant. After an Emden-Fowler transformation,
the inequality can be rewritten as an optimal inequality of logarithmic Sobolev
type on the cylinder. Explicit expressions of the sharp constant, as well as
minimizers, are established in the radial case. However, when no symmetry is
imposed, the sharp constants are not achieved among radial functions, in some
range of the parameters.
