Show a given analytic function is constant

Suppose that ff is analytic on some region R∈CR\in\mathbb{C}. If Im(f)(f) = k⋅k\cdotRe(f)(f) for some nonzero constant k∈Ck\in\mathbb{C}, then show that ff is constant on RR.

I know that if f′(z)=0f'(z)=0 for all zz in some region RR, then f(z)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 kk had better be a real number, since both Im(f)\text{Im}(f) and Re(f)\text{Re}(f) are real-valued.
– Ted Shifrin
Apr 24 at 2:41


1 Answer


If ff is analytic, so is the real-valued function

g(z)=f(z)1+ikg(z) = \frac{f(z)}{1 + ik}

Now there’s a straightforward application of the Cauchy-Riemann equations to conclude that a real-valued analytic function is constant.

Notice that there is one value of kk for which this doesn’t work.