  Both direct and reverse log-Sobolev inequalities, relating the Shannon
entropy with a $\mu$-deformed energy, are shown to hold in a family of
$\mu$-deformed Segal-Bargmann spaces. This shows that the $\mu$-deformed energy
of a state is finite if and only if its Shannon entropy is finite. The direct
inequality is a new result, while the reverse inequality has already been shown
by the authors but using different methods. Next the $\mu$-deformed energy of a
state is shown to be finite if and only if its Dirichlet form energy is finite.
This leads to both direct and reverse log-Sobolev inequalities that relate the
Shannon entropy with the Dirichlet energy. We obtain that the Dirichlet energy
of a state is finite if and only if its Shannon entropy is finite. The main
method used here is based on a study of the reproducing kernel function of
these spaces and the associated integral kernel transform.
