Zygmund theorem for harmonic quasiregular mappings

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

$$\begin{aligned} \Vert f\Vert _{1}\le C(K)(1+\Vert \textrm{Re}\,(f)\log ^+ |\textrm{Re}\, f|\Vert _1), \end{aligned}$$

provided that \(\textrm{Im}\,f(0)=0\). 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

$$\begin{aligned} \Vert f\Vert _{1}-|f(0)|\le (n-1)K^2(\Vert f_1\log f_1\Vert _1- f_1(0)\log f_1(0)), \end{aligned}$$

when \(K\rightarrow 1\), provided that \(f_1\) is positive.