Analytic complex functions

Question

The question at hand is:

Let $f(z)$ be an analytic function on a connected open set $D$. If there are two constants $c_1$, $c_2$ $∈C$, not all zero,such that $c_1 f(z)+c_2\overline{f(z)}=0 $, $\forall z∈D$,then $f(z)$is a constant on $D$.

I think the trick is to use Cauchy-Riemann equations, however I did not come far.

Now I have been studying complex analysis for about a month now, and find it fascinating, however I have been struggling intuitively, which is kind of encapsulated in the question because it looks like in can easily solved but I do not know what property of holomorphic functions should I use. I am asking not so much for a solution, more like a hint/advice how I should approach exercises that rely on function being holomorphic.

If $c_2=0$ you get that $f\equiv 0$. Assume that $c_2\neq0$ applying $\frac{\partial}{\partial\overline{z}}$ on both sides and using Cauchy-Riemann (i.e. that $\frac{\partial}{\partial\overline{z}}f=0$) we get that $c_2\overline{f'(z)}=0$. Therefore, the derivative of $f$ is zero. — plop, May 07 '21 at 20:41
If $c_1\ne 0$, apply $\frac\partial{\partial z}$, otherwise apply $\frac\partial{\partial \overline z}$. — Hagen von Eitzen, May 07 '21 at 20:42
Does this answer your question? How can I understand and prove the "sum and difference formulas" in trigonometry? — GEdgar, May 07 '21 at 21:05

GEdgar · Accepted Answer · 2021-05-07T21:01:48.160

Suppose $c_1 \ne 0$. (The case $c_2 \ne 0$ is similar.) $$ c_1 f(z)+c_2\overline{f(z)}=0 $$ Multiply by $f(z)$, $$ c_1 f(z)^2+c_2|f(z)|^2=0 \\ f(z)^2=-\frac{c_2}{c_1}|f(z)|^2 $$ Thus $f(z)^2$ is analytic, all of its values are in $$ \left\{-\frac{c_2}{c_1}\lambda : \lambda \in \mathbb R\right\} $$ a line through $0$, so by the open mapping theorem, $f(z)^2$ is constant. Since $D$ is connected, we conclude that $f(z)$ is constant.

Analytic complex functions

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