  We prove a new family of inequalities involving squares of random variables
belonging to the Wiener chaos associated with a given Gaussian field. Our
result provides a substantial generalisation, as well as a new analytical
proof, of an estimate by Frenkel (2007), and also constitute a natural real
counterpart to an inequality established by Arias-de-Reyna (1998) in the
framework of complex Gaussian vectors. We further show that our estimates can
be used to deduce new lower bounds on homogeneous polynomials, thus partially
improving results by Pinasco (2012), as well as to obtain a novel probabilistic
representation of the remainder in Hadamard inequality of matrix analysis.
