  The g-function was introduced by Affleck and Ludwig as a measure of the
ground state degeneracy of a conformal boundary condition. We consider this
function for perturbations of the conformal Yang-Lee model by bulk and boundary
fields using conformal perturbation theory, the truncated conformal space
approach and the thermodynamic Bethe Ansatz (TBA). We find that the TBA
equations derived by LeClair et al describe the massless boundary flows, up to
an overall constant, but are incorrect when one considers a simultaneous bulk
perturbation; however the TBA equations do correctly give the `non-universal'
linear term in the massive case, and the ratio of g-functions for different
boundary conditions is also correctly produced. This ratio is related to the
Y-system of the Yang-Lee model and by comparing the perturbative expansions of
the Y-system and of the g-functions we obtain the exact relation between the UV
and IR parameters of the massless perturbed boundary model.
