Invariance of a quasi-arithmetic mean with respect to a system of generalized Bajraktarević means

Let a positive integer k≥2k≥2 and an interval I⊂RI⊂R be fixed. For a continuous strictly monotonic function f:I→Rf:I→R, and arbitrary continuous function g1,…,gk:I→(0,∞)g1,…,gk:I→(0,∞), we define a system of means B[g1,…,gk][f;σki]:Ik→I for i∈{0,1,…,k−1}i∈{0,1,…,k−1}, where σki is the iith iterate of a cycle permutation of the variables. These means generalize the Bajraktarević means. We show that the quasi-arithmetic mean of the generator ff is invariant with respect to the mean-type mapping of this system. The effective formula for the limit of the iterates of these mean-type mappings is given. An application in solving a functional equation is presented.