Riesz and Kolmogorov inequality for harmonic quasiregular mappings

Let K⩾1 and p∈(1,2]. We obtain an asymptotically sharp constant c(K,p) as K→1 in the inequality‖ℑf‖p≤c(K,p)‖ℜf‖p, where f∈hp is a K-quasiregular harmonic mapping in the unit disk belonging to the Hardy space hp. This result holds under the conditions arg⁡(f(0))∈(−π2p,π2p) and f(D)∩(−∞,0)=∅. Our findings improve a recent result by Liu and Zhu [12]. Additionally, we extend this result to K-quasiregular harmonic mappings in the unit ball in Rn. Finally, we consider the Kolmogorov theorem for quasiregular harmonic mappings in the plane.