Maximum principle for proper subset of the boundary of positive measure (Rudin, Function Theory on the Unit Ball of $\mathbb C^n$)

Question

Suppose that $\phi$ is an holomorphic bounded function on the unit ball of $\mathbb C$, such that $\phi(re^{i\theta})$ tends to $0$ as $r$ tends to $1$ for $\theta$ in a positive-measure set $\subseteq[0,2\pi]$.

I have to prove that $\phi(z)=0 \forall z$. Can I apply some refined form of the maximum principle?

The problem is in Rudin, Function theory on the unit ball of $\mathbb C^n$, proof of Lemma 15.2.3.

Thanks in advance.

Answer 1

In my copy of Rudin's Real and Complex analysis it is Theorem 15.19, and I think people call it Jensen's inequality.

If $f$ is analytic in the unit disc, bounded and $f^*$ are its radial limits then $$\log|f(0)|\leq\frac{1}{2\pi}\int_{-\pi}^{\pi}\log|f^*(e^{it})|dt$$ In particular, if the function is not identically zero then the right-hand side is not $-\infty$.

If $\phi$ is not identically 0, then the integral $\frac{1}{2\pi}\int_{-\pi}^{\pi}\log|\phi(e^{it})|dt\geq |\log\phi(0)|$. In the inequality $\phi(e^{it})$ are really the radial limits of $\phi$. Therefore the left-hard side is not $-\infty$. But is the zero set of the boundary values has positive measure then that integral is $-\infty$.