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 $$\mathrm{\parallel }f{\mathrm{\parallel }}_1\le C(K)(1+\mathrm{\parallel }\mathrm{R}\mathrm{e}(f){\log }^{+}\mathrm{\mid }\mathrm{R}\mathrm{e}f\mathrm{\mid }{\mathrm{\parallel }}_1),$$ provided that $\mathrm{I}\mathrm{m}f(0)=0$. 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 $$\mathrm{\parallel }f{\mathrm{\parallel }}_1-\mathrm{\mid }f(0)\mathrm{\mid }\le (n-1)K^2(\mathrm{\parallel }{f}_1\log f_1{\mathrm{\parallel }}_1-f_1(0)\log f_1(0)),$$ when $K\to 1$, provided that $f_1$ is positive.