Zygmund theorem for harmonic quasiregular mappings

Abstract: Let $K\ge 1$. We prove Zygmund theorem for $K-$quasiregular harmonic mappings in the unit disk $\mathbb{D}$ in the complex plane by providing a constant $C(K)$ in the inequality $$|f|{1}\le C(K)(1+|\mathrm{Re}\,(f)\log^+ |\mathrm{Re}\, f||_1),$$ provided that $\mathrm{Im}\,f(0)=0$. Moreover for a quasiregular harmonic mapping $f=(f_1,\dots, f_n)$ defined in the unit ball $\mathbb{B}\subset \mathbb{R}^n$, we prove the asymptotically sharp inequality $$|f|{1}-|f(0)|\le (n-1)K^2(|f_1\log f_1|_1- f_1(0)\log f_1(0)),$$ when $K\to 1$, provided that $f_1$ is positive.