# Show a given analytic function is constant

Suppose that $f$ is analytic on some region $R\in\mathbb{C}$. If Im$(f)$ = $k\cdot$Re$(f)$ for some nonzero constant $k\in\mathbb{C}$, then show that $f$ is constant on $R$.

I know that if $f'(z)=0$ for all $z$ in some region $R$, then $f(z)$ is constant. However, I'm not sure how to apply this to the question at hand. I also think the Cauchy-Riemann equations could be helpful, but again, I'm not sure how to apply them to the question.

Any guidance would be greatly appreciated. Thank you!

• The constant $k$ had better be a real number, since both $\text{Im}(f)$ and $\text{Re}(f)$ are real-valued. Apr 24 '16 at 2:41

If $f$ is analytic, so is the real-valued function
$$g(z) = \frac{f(z)}{1 + ik}$$
Notice that there is one value of $k$ for which this doesn't work.