Let $f(z)$ be a holomorphic function $\mathbb{D}^* := \{z \in \mathbb{C} : 0 < |z| < 1\}$ with an essential singularity at $0$. Denote $l(r)$ as the length of the curve $\gamma_r : [0,2\pi] \to \mathbb{C}$ given by $\gamma_r(\theta) = f(re^{i\theta})$. Show that for every real number $\alpha$, we have: $$ \lim_{r \to 0} r^\alpha l(r) = +\infty $$
My first observation is that it suffices to prove for all $\alpha \in \mathbb{Z}$, as the rest of the real numbers follows by squeeze theorem. I then observe that we can reduce it further to the case of $\alpha = 1$. Indeed, the length of the curve is given by: $$ l(r) = \int_0^{2\pi} |\gamma_r(\theta)| \; \mathrm{d}\theta = \int_0^{2\pi} |f(re^{i\theta})| \; \mathrm{d}\theta $$ Suppose we have proven the case for $\alpha = 1$. For any $n \in \mathbb{Z}$, the function $g_n(z) = z^nf(z)$ is clearly holomorphic in $\mathbb{D}^*$. Furthermore, $0$ is an essential singularity for $g_n(z)$, as writing $f(z)$ in it's Laurent expansion yields: $$ g_n(z) = z^n\sum_{k = -\infty}^\infty a_kz^k = \sum_{k = -\infty}^\infty a_kz^{n + k} $$ It's clear that the Laurent expansion of $g_n(z)$ above has infinitely many non-zero coefficient for negative powers. Thus, for the case of $g_n(z)$: $$ \int_0^{2\pi} |\gamma_r(\theta)| \; \mathrm{d}\theta = \int_0^{2\pi} |r^ne^{ni\theta}f(re^{i\theta})| \; \mathrm{d}\theta = r^n \int_0^{2\pi} |f(re^{i\theta})| \; \mathrm{d}\theta = r^nl(r) \to +\infty $$ However, I'm not sure how to proceed with the $n = 1$ case. We probably should apply Casorati-Weierstrass Theorem, in which in particular $\{f(re^{i\theta})\}$ is unbounded as $r \to 0$. However, this assertion is clearly insufficient to conclude that $l(r) \to 0$, as it's possible that the "length" of the circle which $f(re^{i\theta})$ becomes arbitrarily large also becomes arbitrarily small.
Any help is appreciated.
