A classical result of Hardy and Littlewood says that if \(f=u+iv\) is analytic in the unit disk \({\mathbb {D}}\) and u is in the harmonic Bergman space \(a^p\) ( \(0<p<\infty \) ), then v is also in \(a^p\) . This complements a celebrated result of M. Riesz on Hardy spaces, which only holds for \(1<p<\infty \) . These results do not extend directly to complex-valued harmonic functions. We prove that the Hardy-Littlewood theorem holds for a harmonic function \(f=u+iv\) if we place the assumption that f is quasiregular in \({\mathbb {D}}\) . This makes further progress on the recent Riesz type theorems for harmonic quasiregular mappings by several authors. Then we consider univalent harmonic mappings in \({\mathbb {D}}\) and study their membership in Bergman spaces. In particular, we produce a non-trivial range of \(p>0\) such that every univalent harmonic function f (and the partial derivatives \(f_\theta ,\, rf_r\) ) is of class \(a^p\) . This result extends nicely to harmonic quasiconformal mappings in \({\mathbb {D}}\) .
