  Let u, v be two harmonic functions in the disk of radius two which have
exactly the same set Z of zeros. We observe that the gradient of \log |u/v| is
bounded in the unit disk by a constant which depends on Z only. In case Z is
empty this goes back to Li-Yau's gradient estimate for positive harmonic
functions. The general boundary Harnack principle gives H\"older estimates on
\log |u/v|.

Analysis of PDEs