  We compute the two-loop effects induced by an anomalous Higgs trilinear
self-coupling in the partial decay width h to gamma Z. The computation is
performed using the anomalous coupling approach working in the unitary gauge
and in a theory in which the anomalous coupling is generated via the addition
to the scalar potential part of the Standard Model Lagrangian of an (in)finite
tower of (Phi^dagger Phi)^n terms. The former computation is automatically
finite while the latter requires the renormalization of the lowest order
contribution. We discuss the renormalization conditions that should be employed
in order to obtain the same result in the two approaches. We find that the h to
gamma Z process is one of the most sensitive mode to an anomalous trilinear
Higgs self-coupling. As a by-product of this work we confirm one of two
different results present in the literature concerning the contribution of an
anomalous Higgs trilinear coupling in the h to gamma gamma decay.
