 Harnack inequality on the unit disk

For a nonnegative harmonic function on the unit disc, the displayed inequalities are sharp. Apply the Poisson integral on the unit disk on radius $R>|z|$, bound its positive kernel between $(R-|z|)/(R+|z|)$ and its reciprocal, and use the mean value property for harmonic functions. Letting $R\uparrow 1$ proves the assertion. Positive Poisson kernels with one boundary pole give equality in the appropriate radial directions.