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 $f=h+\bar{g}$ 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 $\zeta =\mu +\bar{\omega }$ such that the real parts of its multipliers satisfy $\text{Re}({\mathrm{\partial }}_{z}h(\mu ))\ge 1$ and $\text{Re}({\mathrm{\partial }}_{z}g(\omega ))\ge 1$. For polynomial harmonic functions, this holds even without super-attracting conditions. We provide explicit examples, visualizations, and discuss problem for transcendental harmonic functions.