Riesz-Fej\'er inequalities for harmonic functions

In this article, we prove the Riesz - Fej'er inequality for complex-valued harmonic functions in the harmonic Hardy space ${\mathbf{h}}^p$ for all $p>1$. The result is sharp for $p\in (1,2]$. Moreover, we prove two variant forms of Riesz-Fej'er inequality for harmonic functions, for the special case $p=2$.