Let $\Sigma_\phi = \{z \in \mathbb C : |\arg(z)|<\phi\}, 0<\phi<\pi$ be a sector and let $f \colon \Sigma_\phi \to \mathbb C$ be holomorphic. If $$ \|f\|_{H^1(\Sigma_\phi)} = \sup_{|\omega|<\phi} \int_0^\infty |f(te^{i\omega})| t^{-1}dt < \infty, $$ then do there exist constants $M, \alpha>0$ such that $|f(z)| \le M \min\{|z|^\alpha, |z|^{-\alpha}\}$ for all $z \in \Sigma_\phi$?
I can show that the converse implication holds, but not this direction, and I'm wondering whether this is also true (perhaps on a(ny) smaller sector $\Sigma_{\omega}$ with $\omega<\phi$).