Why $\|f\|_p=u_f(0)^\frac{1}{p}$?

Question

Suppose that $f \in H^p$, where $0<p<\infty$. Let $u_f$ be the least harmonic majorant of $|f|^p$. Why is it true that \begin{equation} \|f\|_p=u_f(0)^\frac{1}{p}? \end{equation} I was hinted to start with the harmonic functions in $D(0;R)$ with boundary values $|f|^p$ for $R<1$ and then let $R \to 1$. However, I don't know how to use this hint. Please help.