  We introduce a countable collection of positivity classes for Hermitian
symmetric functions on a complex manifold, and establish their basic
properties. We study a related notion of stability. The first main result shows
that, if the underlying matrix of coefficients of an entire Hermitian symmetric
function has at most k positive eigenvalues, then it can lie in the k-th
positivity class only if it is a squared norm. We establish a similar result
for Hermitian symmetric functions on the total space of a holomorphic line
bundle. Finally we study the positivity classes for a natural one-parameter
family of Hermitian metrics on a power of the universal bundle over complex
projective space; we obtain sharp information about the parameter values in
order to be in the k-th class. The paper closes with some additional
information about the case when k is 2, where a nonlinear version of the
Cauchy-Schwarz inequality arises.
