In [1] Bazilevic` proved a remarkable averaging theorem for univalent functions. It is the aim of this article to show that Bazilevic`'s theorem may be proved, in a slightly weaker form, for the more general class of areally meanp‐valent (a.m.p.v.) functions. Bazilevic`'s method used the univalence property very strongly, since he needed the Grunsky inequalities. Thus a different proof is essential for the generalization. Our approach is close in nature to the one appearing in Hayman's paper [4].