A gradient estimate for harmonic functions sharing the same zeros

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|.