  Let H be any complex inner product space with inner product <, >. We say that
f : C -->C is Hermitian positive definite on H if the matrix
$$(f(<z^r,z^s>))_{r,s=1}^n \eqno(*)$$ is Hermitian positive definite for all
choice of z^1,...,z^n in H, all n. It is strictly Hermitian positive definite
if the matrix (*) is also non-singular for any choice of distinct z^1,...,z^n
in H. In this article we prove that if dim H >= 3, then f is Hermitian positive
definite on H if and only if $$f(z) = \sum_{k,m =0}^\infty b_{k,m} z^k \oz^m
\eqno(**)$$ where \oz is the conjugate of z, b_{k,m}>= 0, all k,m in Z_+, and
the series converges for all z in C. We also prove that f of the form (**) is
strictly Hermitian positive definite on any H if and only if the set $$J={(k,
m) : b_{k,m}> 0}$$ is such that (0,0) is in J, and every arithmetic sequence in
Z intersects the values {k-m : (k,m)\in J} an infinite number of times.
