Real non-attractive fixed point conjecture for complex harmonic functions

We prove the real non-attractive fixed point conjecture for complex polynomial and rational harmonic functions. A harmonic function $f=h+\overline{g}$ is polynomial (rational) if both $h$ and $g$ are polynomials (rational functions) of degree at least 2. We show that every such function with a super-attracting fixed point has a $\mathfrak{h}$-fixed point $\zeta=\mu+\overline{\omega}$ such that the real parts of its multipliers satisfy $\text{Re}(\partial_z h(\mu)) \geq 1$ and $\text{Re}(\partial_z g(\omega)) \geq 1$. For polynomial harmonic functions, this holds even without super-attracting conditions. We provide explicit examples, visualizations, and discuss problem for transcendental harmonic functions.