2025/07/24 by Vaseem, Mohd
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.18414
We prove the real non-attractive fixed point conjecture for complex polynomial and rational harmonic functions. A harmonic function f=h+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 \mathfrakh-fixed point ζ=μ+ω such that the real parts of its multipliers satisfy Re(∂z h(μ)) ≥ 1 and Re(∂z g(ω)) ≥ 1. For polynomial harmonic functions, this holds even without super-attracting conditions. We provide explicit examples, visualizations, and discuss problem for transcendental harmonic functions.