AbstractIt is known that if a bivariate mean K is symmetric, continuous and strictly increasing in each variable, then for every mean M there is a unique mean $$N\,$$
N
such that K is invariant with respect to the mean-type mapping $$\left( M,N\right) ,$$
M
,
N
,
which means that $$K\circ \left( M,N\right) =K$$
K
∘
M
,
N
=
K
and N is called a K-complementary mean for M (Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations.