Consider the following theorem (see theorem 7.17 of [W. Rudin. Priniciples of Mathematical Analysis, Mcgraw-Hilly, 1976.])
Theorem. Suppose $\{f_n\}$ is sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point $x_0$ on $[a,b]$. If $\{f'_n\}$ converges uniformly on $[a,b]$, then $\{f_n\}$ converges uniformly on $[a,b]$ to a function $f$ and
\begin{align*} f'(x)=\lim_{n\to\infty}f'_n(x),\;\;a\leq x \leq b. \end{align*}
>I know that the Theorem is stated for real case, but in my research I dealt to complex case and Now my question is :
Does there exist the above (or similar) theorem for complex case? (i.e., Suppose $\{f_n\}$ is sequence of complex functions, differentiable on an open domain $D$ and such that $\{f_n(z_0)\}$ converges for some point $z_0$ on an open domain $D$. If $\{f'_n\}$ converges uniformly on an open domain $D$, then $\{f_n\}$ converges uniformly on an open domain $D$ to a function $f$ and $f'(z)=\lim_{n\to\infty}f'_n(z),\;\;z\in D.$)? If yes, what is refrences. Anyone can help me? Thanks in advance.
Here is one version of the theorem for the complex case
Let $D$ be a convex open set and $f_n,f$ be holomorphic functions on $D$. If $f_n'$ converges uniformly on compact subsets of $D$ and $f_n(z_0) \to f(z_0)$ for some $z_0$ then there is a holomorphic function $h$ such that $f_n \to h$ uniformly on compact subsets and $\lim f_n'(z)=h'(z)$.
Proof: $f_n(z)=f_n(z_0)+\int_{[z_0,z]}f_n'(w)dw$ where $[z_0,z]$ denotes the line segment from $z_0$ to $z$. if $g$ is the limit of $f_n'$ the we see that $f_n$ converges uniformly on compact stes to a funtion $h$ such that $h(z)=f(z_0)+\int_{[z_0,z]}g'(w)dw$. This implies that $h'=g$.