The following is taken from Rudin's Principles of Mathematical Analysis; it is Theorem 7.17 with some extra assumptions made based on a remark at the end of the proof of the theorem.
Let $f_n,g$ be functions defined on an interval $[a,b]$. Assume that $f_n$ are differentiable with continuous derivatives. Assume also that $f_n' \rightarrow g$ uniformly and $f_n \rightarrow f$ pointwise in $[a,b]$ for some function $f$ (or more generally, assume that $f_n(x_o)$ is convergent for some $x_o \in [a,b]$). Then, $f_n \rightarrow f$ uniformly, $f$ is differentiable with continuous derivative and $f'=g$.
Does a similar result hold for holomorphic functions on an open set? Since Rudin speaks of intervals, should such a result be valid for connected or convex sets only, instead of open sets in general? Namely, if $\Omega$ is a domain and $f_n$ are holomorphic functions on $\Omega$, does uniform convergence of $f_n′$ on the whole domain (not uniform convergence on compacts subsets of the domain!) plus pointwise convergence of $f_n$, imply uniform convergence of $f_n$ on the whole domain?
Best Answer
Yes. Is an easy consequence of Morera's theorem and uniform convergence on compact sets sets is sufficient. For your exact hypothesis a connected domain is required.