If u is a positive harmonic function on U, the open unit disc, such that u(0)=1. Find the upper and lower bounds for u(1/2).
Note that u is only harmonic 'inside' U.
Can we estimate u(1/2) only by using the concepts of Poisson integral and Harnack inequality? I know that u should be given by a Poisson integral of a holomorphic function defined on the boundary, which I guess to be u itself. But we have no information about u on the boundary. Could someone help??