The Cauchy Transform, Compact Convergence on the Disk, and Weak Convergence in $L^{p}$

Question

Let $p\in\left[1,\infty\right)$, and let $\left\{ f_{m}\right\} _{\geq1}$ be a sequence of holomorphic functions on the open unit disk $\mathbb{D}$ that converges uniformly on compact subsets of $\mathbb{D}$ to a holomorphic limit $f:\mathbb{D}\rightarrow\mathbb{C}$. Suppose that there is a sequence $\left\{ \phi_{m}\right\} _{m\geq1}\subseteq L^{p}\left(\mathbb{R}/\mathbb{Z}\right)$ so that:$$f_{m}\left(z\right)=\int_{0}^{1}\frac{\phi_{m}\left(t\right)dt}{1-e^{-2\pi it}z},\textrm{ }\forall\left|z\right|<1,\textrm{ }\forall m\geq1$$ Is it then true that the $\phi_{m}$s converge weakly in $L^{p}\left(\mathbb{R}/\mathbb{Z}\right)$ to a limit $\phi\in L^{p}\left(\mathbb{R}/\mathbb{Z}\right)$, and that: $$f\left(z\right)=\int_{0}^{1}\frac{\phi\left(t\right)dt}{1-e^{-2\pi it}z},\textrm{ }\forall\left|z\right|<1$$