Riesz and Kolmogorov inequality for harmonic quasiregular mappings

Let K 1and p.(1, 2]. We obtain an asymptotically sharp constant c( K, p) as K. 1in the inequality f p = c( K, p) f p, where f. hpis a K-quasiregular harmonic mapping in the unit disk belonging to the Hardy space hp. This result holds under the conditions arg(f(0)). - p2p, p2p and f(D) n(-8, 0) =O. 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. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.