  Deformed Harmonic Oscillator Algebras are generated by four operators, two
mutually adjoint $a$ and $a^\dagger$, and two self-adjoint $N$ and the unity
$1$ such as:
  $[a,N] = a, [a^\dagger, N]= -a^\dagger, a^\dagger a = \psi(N)$ and
$aa^\dagger =\psi(N+1)$.
  The Bargmann Hilbert space is defined as a space of functions, holomorphic in
a ring of the complex plane, equipped with a scalar product involving a true
integral. In a Bargmann representation, the operators of a Deformed Harmonic
Oscillator Algebra act on a Bargmann Hilbert space and the creation (or the
annihilation operator) is the multiplication by $z$.
  We discuss the conditions of existence of Deformed Harmonic Oscillator
Algebras assumed to admit a given Bargmann representation.
