Let $f(z),z^5\bar{f}(z)$ be entire functions on $\mathbb{C}$. Show that $f$ is constant.
I tried using Cauchy-Riemann quations in their polar form in order to find out the derivaties are zero and I got that $$\begin{cases}u_r=\frac 1 r v_\theta\\v_r=-\frac 1 r u_\theta\\ \frac{\partial}{\partial r}\Re(r^5 cis(\theta)f(r,\theta))=\frac{1}{r}\frac{\partial}{\partial \theta}\Im(r^5 cis(\theta)f(r,\theta))\\\frac{\partial}{\partial r}\Im(r^5 cis(\theta)f(r,\theta))=-\frac{1}{r}\frac{\partial}{\partial \theta}\Re(r^5 cis(\theta)f(r,\theta))\end{cases}$$ which is a complex system of PDEs. I also thought using Liouville's theorem but I don't know how to prove that $f$ is a bounded function. Can I get any hint?
If $f(z)$ is entire, so is $z^5\, f(z)$. Hence $g(z)=z^5(f(z)+\overline f(z))$ is entire and $g(z)/z^5$ is a holomorphic real valued function in $\Bbb C\setminus \{0\}$.