Note on real and imaginary parts of harmonic quasiregular mappings

Abstract If $f=u+iv$ is analytic in the unit disk ${\mathbb D}$ , it is known that the integral means $M_p(r,u)$ and $M_p(r,v)$ have the same order of growth. This is false if f is a (complex-valued) harmonic function. However, we prove that the same principle holds if we assume, in addition, that f is K -quasiregular in ${\mathbb D}$ . The case $0<p<1$ is particularly interesting, and is an extension of the recent Riesz-type theorems for harmonic quasiregular mappings by several authors. Further, we proceed to show that the real and imaginary parts of a harmonic quasiregular mapping have the same degree of smoothness on the boundary.