  We identify measures arising in the representations of products of
generalized Stieltjes transforms as generalized Stieltjes transforms, provide
optimal estimates for the size of those measures, and address a similar issue
for generalized Cauchy transforms. In the latter case, in two particular
settings, we give criteria ensuring that the measures are positive. On this
way, we also obtain new, applicable conditions for representability of
functions as generalized Stieltjes transforms, thus providing a partial answer
to a problem posed by Sokal and shedding a light at spectral multipliers
emerged recently in probabilistic studies. As a byproduct of our approach, we
improve several known results on Stieltjes and Hilbert transforms.
