Iterations of mean-type mappings and invariant means. (English) Zbl 0954.26015

Let \(I\) be a real interval, \(M_k:I^p\to I\) be continuous functions and mean values, that is, \[ \min(x_1,\dots{},x_p)\leq M_k(x_1,\dots{},x_p)\leq \max(x_1,\dots{},x_p) \tag{M} \] for all \(x_j\in I\) \((j=1,\dots{},p\), \(k=1,2,\dots{})\) with all but (at most) one \(M_k\) a strict mean (i.e., \(\leq\) in (M) is replaced by \(<\) if the \(x_1,\dots{},x_p\) are not all equal). Define further \(M_{k,1}=M_k,\) \[ M_{k,n+1}(x_1,\dots{},x_p)=M_k(M_{1,n}(x_1,\dots{},x_p),\dots{}, M_{k,n}(x_1,\dots{},x_p)) \] \((k=1,\dots{},p\); \(n=1,2,\dots{}).\) In the case where \(p=2\) and \(M_1,M_2\) is the arithmetic and geometric mean, respectively, this algorithm leads, of course, to the medium arithmetico-geometricum of Gauss. Here is a couple of other references for this type of iteration of means: I. Fenyö [Acta Univ. Szeged., Acta Sci. Math. 13, 36-42 (1949; Zbl 0034.32701)] and J. M. Borwein and P. B. Borwein [Am. Math. Mon. 94, 519-522 (1987; Zbl 0628.26003)]. The author offers as main result that under these conditions the sequences \(\{M_{k,n}\}_{n=1}^\infty\) have a unique common (pointwise) limit \(K,\) which is a continuous mean value and satisfies (obviously) \[ K(x_1,\dots{},x_p)=K(M_1(x_1,\dots{},x_p),\dots{}, M_p(x_1,\dots{},x_p)). \] Remarks and examples follow.


26E60 Means
39B12 Iteration theory, iterative and composite equations
26A18 Iteration of real functions in one variable
54H25 Fixed-point and coincidence theorems (topological aspects)
26B40 Representation and superposition of functions
39B22 Functional equations for real functions
40A30 Convergence and divergence of series and sequences of functions