Holomorphic functions of certain blow up at origin

Question

Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\limits_{r\to 0^+} f(re^{i\theta})$ exists and is equal to a constant for all $\theta \in (-\frac{\pi}{4},\frac{\pi}{4}).$ Does there exist a constant $C>0$ independent of $f$ such that $$ \left|\lim\limits_{r\to 0}f(re^{i\theta})\right| \leq C \|f\|_{H^1(\partial D)}?$$

Answer 1

The answer is no, consider $f_\epsilon(z) = \frac{e^{-\frac{\epsilon}{z}}-1}{100}$. Then for all $\epsilon < 1$ the function satisfies the required bound, the limit is $\frac{1}{100}$ regardless of $\epsilon$ but the $H^1$-norm is at most $\epsilon$ which can be arbitrarily small.