## 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.

### MSC:

 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

### Citations:

Zbl 0034.32701; Zbl 0628.26003