Riesz and Kolmogorov inequality for harmonic quasiregular mappings

Let $K\ge 1$ and $p\in (1,2]$. We obtain asymptotically sharp constant $c(K,p)$, when $K\to 1$ in the inequality $$\mathrm{\parallel }\mathrm{\Im }f{\mathrm{\parallel }}_p\le c(K,p)\mathrm{\parallel }\mathrm{\Re }(f){\mathrm{\parallel }}_p$$ where $f\in {\mathbf{h}}^p$ is a $K-$quasiregular harmonic mapping in the unit disk belonging to the Hardy space ${\mathbf{h}}^p$, under the conditions $\arg (f(0))\in (-\pi \mathrm{/}(2p),\pi \mathrm{/}(2p))$ and $f(\mathbb{D})\cap (-\mathrm{\infty },0)=\mathrm{\varnothing }$. The paper improves a recent result by Liu and Zhu in \cite{aimzhu}. We also extend this result for the quasiregular harmonic mappings in the unit ball in ${\mathbb{R}}^n$. We also extend Kolmogorov theorem for quasiregular harmonic mappings in the plane.