Given an analytic function \(f=u+iv\) in the unit disk \(\mathbb {D}\) , Zygmund’s theorem gives the minimal growth restriction on u which ensures that v is in the Hardy space \(h^1\) . This need not be true if f is a complex-valued harmonic function. However, we prove that Zygmund’s theorem holds if f is a harmonic K-quasiregular mapping in \(\mathbb {D}\) . Our work makes further progress on the recent Riesz-type theorem of Liu and Zhu (Adv. Math., 2023), and the Kolmogorov-type theorem of Kalaj (J. Math. Anal. Appl., 2025), for harmonic quasiregular mappings. We also obtain a partial converse, thus showing that the proposed growth condition is the best possible. Furthermore, as an application of the classical conjugate function theorems, we establish a harmonic analogue of a well-known result of Hardy and Littlewood.
