  We prove that a positive function on the unit disk admits a harmonic majorant
if and only if a certain logarithmic Lipschitz upper envelope of it (relevant
because of the Harnack inequality) admits a superharmonic majorant. We discuss
the logarithmic Lipschitz regularity of this superharmonic majorant, and show
that in general it cannot be better than that of the Poisson kernel.
  We provide examples to show that mere superharmonicity of the data does not
help with the problem of existence of a harmonic majorant.
