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

Let a positive integer k ≥ 2 and an interval I ⊂ R be fixed. For a continuous strictly monotonic function f : I → R , and arbitrary continuous function g 1 , … , g k : I → ( 0 , ∞ ) , we define a system of means B [ g 1 , … , g k ] [ f ; σ k i ] : I k → I  for i ∈ { 0 , 1 , … , k − 1 } , where σ k i is the i th iterate of a cycle permutation of the variables. These means generalize the Bajraktarević means. We show that the quasi-arithmetic mean of the generator f 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.