Existence of holomorphic function with given conditions

Question

I would like to answer this question:

Let $\mathbb{E}:= \{z \in \mathbb{C}: |z| < 1\}$. Does there exist a holomorphic function $f: \mathbb{E} \to \mathbb{C}$ with $ f(0) = 2i$ and $\lim_{|z| \to 1}|f(z)| = 1$?

If we would replace $\lim_{|z| \to 1}|f(z)| = 1$ with the condition $|f(z)| = 1$ for all $z \in \mathbb{E}$, then I would know that the answer would be negative (identity theorem). Here I am unsure about whether or not the condition $\lim_{|z| \to 1}|f(z)| = 1$ changes something in regard to use the identity theorem.

I am thankful for some clarification.

Answer 1

Since $\lim_{|z| \to 1}|f(z)| = 1$, there is an $r \in (0, 1)$ such that $|f(z)| \le 1.5$ for $r \le |z| < 1$.

The maximum modulus principle then implies that $|f(z)| \le 1.5$ for $|z| \le r$, so that $f(0) = 2i$ is not possible.

Remark: The same argument shows that if $f$ is holomorphic in the unit disk with $\lim_{|z| \to 1}|f(z)| = 1$ then $|f(z)| \le 1$ for all $z$.