The question is to find all differentiable functions $f:u+iv:\Omega \rightarrow \mathbb{C}$, $\Omega$ an open connected set, such that $u=3v$ and $f(0)=3+i$.
I know Cauchy-Riemann conditions have to be satisfied in order for the function to be differentiable so I wrote $u_x =v_y$ and $u_y=-v_x$ (where $u_x$ is partially derived from $u$ with respect to $x$ and so on). Additionally, from $u=3v$, I think I can conclude $u_x=3v_x$ and $u_y=3v_y$. Therefore, I also have $v_y=3v_x$ and $-v_x=3v_y$. I believe this determines $v$ to be a constant function with respect to both $x$ and $y$. So from $f(0)=3+i$ I concluded $v(0)=1$ and $u(0)=3$ and since $v$ is constant it means $v(x+iy)=1$. So then $u(x+iy)=3$. However, I don't know whether this is correct or if I'm missing something since I have no other ideas.