Suppose that $f$ is a holomorphic function on a domain $D$ in $\mathbb{C}^n$, $\partial D$ is smooth, and $f$ is $C^1$ on $\partial D$. Then, the Bochner-Martinelli formula states that $f(z) = \int_{\partial D} f(\zeta) \omega(\zeta, z)$, where $\omega(\zeta, z)$ is the Bochner-Martinelli kernel. I am wondering if a $C^1$ function satisfying this formula is necessarily holomorphic.

This question arose when I was trying to prove that if a sequence of $C^1$ functions $f_n:D\rightarrow\mathbb{C}$ satisfies $f_n\rightarrow f$ uniformly and $\partial\bar{f}_n\rightarrow0$ uniformly, then $f$ is holomorphic. Is this statement correct?