Actually it is rudin's exercise 11.13. Let $u$ is positive harmonic on $U$, unit disc and $u(0)=1$. How can large $u(1/2)$ be?
What I tried is this; By theorem 11.30 that every positive harmonic function has its unique positive borel measure $\mu \in M(\mathbb{T})$ such that $u = P[d\mu]$. Also by mean value property, $u(0) = \frac{1}{2\pi} \int_{-\pi}^{\pi}u(\frac{e^{it}}{2})dt$. What can I do more on that?
By Herglotz representation theorem we know that $$u(re^{i\theta})=\int_0^{2\pi} \frac{1-r^2}{1-2r\cos(\theta-\phi)+r^2}d\mu(\phi),$$ for some probability measure $\mu$, and so $$u(1/2)=\int_0^{2\pi} \frac{3}{4-4\cos(\phi)+1}d\mu(\phi).$$ Now, since $-1\leq \cos(\phi)\leq 1$, this quantity is at least $\frac{1}{3}$ and at most $3$, and both of these extremal values can be realized by taking $\mu$ to be the point mass at $\pi$ and $0$ respectively.