Boundary behavior of $f\in Hol (\mathbb{D})$

Question

Let $\text{Hol}(\mathbb{D})$ denotes the set of holomorphic functions on the open unit disc $\mathbb{D}$. Prove that

1. There is no $f\in \text{Hol}(\mathbb{D})$, such that $$\lim_{|z|\to 1^-}|f(z)|=+\infty.$$
2. There exists $f\in \text{Hol}(\mathbb{D}),$ $f\not\equiv 0,$ and a dense subset $E$ of $\mathbb{T}$, such that $$\lim_{r\to 1^-} |f(rw)|=0,$$ for $w\in E$.

---

Here are my thoughts.

1.For the first problem, if $f$ has no zero. Consider $\textrm{log}|f|$, this is a harmonic function on $\mathbb{D}$. So by mean-value principal, $$\frac{1}{2\pi r}\int_{|z|=r}log|f|=log|f(0)|< +\infty.$$ Let $r\to 1^-$, a contradiction. If $f$ has zero, from the condition, $f$ has only finite many zero. Write $f=Bg$, where $B$ is a Blaschke product. Apply above discussion to $g$. Is this argument right?

2.For the second problem. I tried to consider the Poisson integral $f=P[\text{d}\mu]$ for some singular complex measure $\mu$ supporting on $\mathbb{T}$. Then by Fatou’s theorem, the radical limit of $f$ is zero almost every where. If the Fourier coefficients of $\mu$ vanish on $n<0$, the $P[\text{d}\mu]$ is holomorphic on $\mathbb{D}$. But unfortunately, Riesz’s theorem claims that every such measure is automatically absolutely continuous. So this idea is not work.

---

Any help or hint? Thank you!

Answer 1

Your proof for 1. is fine.

You might be interested in this more powerful result: Let $A$ be an arc on $\partial \mathbb D$ of positive length. Then there is no holomorphic $f$ on $\mathbb D$ such that

$$\lim_{r\to 1^-} |f(re^{it})| = \infty$$

for all $e^{it}\in A.$

For 2. we can do this: Let $\zeta_1, \zeta_2, \dots $ be a dense subset of $\partial \mathbb D.$ For each $n,$ let $\mu_n$ be the point mass at $\zeta_n$ with $\mu_n(\zeta_n)=2^{-n}.$ Define $u_n$ to be the Poisson integral of $\mu_n.$ Then for each $n,$ $u_n$ is positive and harmonic on $\mathbb D$ and

$$\lim_{r\to 1^-} u_n(r\zeta_n) =\infty.$$

Define $u=\sum_{n=1}^{\infty} u_n.$ Then $u$ is positive and harmonic on $\mathbb D,$ with radial limit $\infty$ at each $\zeta_n.$ Letting $v$ be a harmonic conjugate of $u,$ we obtain $e^{-(u+iv)},$ a bounded holomorphic function on $\mathbb D,$ with no zero in $\mathbb D,$ having radial limit $0$ at each $\zeta_n.$