On the quasi-arithmetic Gauss-type iteration

For a sequence of continuous, monotone functions f1,…,fn:I→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1,\ldots ,f_n :I \rightarrow \mathbb {R}$$\end{document} (I is an interval) we define the mapping M:In→In\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M :I^n \rightarrow I^n$$\end{document} as a Cartesian product of quasi-arithmetic means generated by fj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_j$$\end{document}-s. It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of In\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^n$$\end{document}. We will prove that whenever all fj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_j$$\end{document}-s are C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}^2$$\end{document} with nowhere vanishing first derivative, then this convergence is quadratic. Furthermore, the limit VarMk+1(v)(VarMk(v))2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\text {Var}\, M^{k+1}(v)}{(\text {Var}\, M^{k}(v))^2}$$\end{document} will be calculated in a nondegenerated case.


Introduction
In 1800 (this year is due to [33]) Gauss introduced the arithmetic-geometric mean as a limit in the following two-term recursion: where x 0 = x and y 0 = y are two positive parameters. Gauss [14, p. 370] proved that both (x k ) ∞ k=1 and (y k ) ∞ k=1 converge to a common limit, which is called the arithmetic-geometric mean of the initial values x 0 and y 0 . Borwein and Borwein [6] extended some earlier ideas [13,22,32] and generalized this iteration to a vector of continuous, strict means of an arbitrary length.
Invariant means in a family of quasi-arithmetic means were studied by many authors, for example Burai [7], Daróczy-Páles [9], Jarczyk [17] and Jarczyk and Matkowski [18]. In fact invariant means were extensively studied in recent years, see for example the papers by Baják-Páles [2][3][4][5], by Daróczy-Páles [8,10,11], by G lazowska [15,16], by Matkowski [23][24][25][26], by Matkowski-Páles [27], and by the author [30]. It is known, [6,Theorem 8.2], that for all twice continuously differentiable, strict means M, N and sequences the difference |x k − y k | tends to zero quadratically for all x 0 = x and y 0 = y. Following [6, section 8.7], we will consider the iteration of multidimensional means. Given a natural number n ∈ N and a vector of means (M 1 , . . . , M n ) defined on a common interval I, let us define the mapping M : Whenever for every i ∈ {1, . . . , N} the limit of its iteration sequence lim k→∞ [M k (a)] i exists and does not depend on i, we call it the invariant mean of (M i ) and denote it by M ⊗ (a). Some authors refers to M ⊗ (a) as Gaussian product. Indeed, M ⊗ can be characterized as a unique mean satisfying the equality M ⊗ • M = M ⊗ (cf. e.g. Matkowski [23]). He also proved that whenever all means are continuous and strict then M ⊗ is a uniquely defined continuous and strict mean.
Some special case is that for some k 0 ∈ N the vector M k0 (a) is constant. Then, for all k ≥ k 0 , we have M k (a) = M k0 (a). In particular each entry of this vector equals M ⊗ (a). If it is the case for some nonconstant vector a, then we will call such an iteration process degenerated. It can be easily verified that under some mild condition regarding the comparability of means an iteration process is never degenerated. Such results are however outside the scope of this paper and are omitted.
Gauss' iteration process in a case when all means are quasi-arithmetic will be of our interest. It was already under investigation in [30]. We are going to continue the research in this area. In particular we will prove the multidimensional counterpart of [6,Theorem 8.2] in a case when all considered means are quasi-arithmetic. Furthermore we will show that, under some conditions, not only the convergence is quadratic, but also the characteristic ratio is closely related to the so-called Arrow-Pratt index.

Quasi-arithmetic means
Quasi-arithmetic means were introduced in a series of nearly simultaneous papers in the early 1930s [12,20,29] as a generalization of the already mentioned family of power means. For a continuous and strictly monotone function f : I → R (I is an interval) and a vector a = (a 1 , a 2 , . . . , a n ) ∈ I n , n ∈ N we define Vol. 92 (2018) On the quasi-arithmetic Gauss-type iteration 1121 It is easy to verify that for I = R + and f = π p , where π p (x) := x p if p = 0 and π 0 (x) := ln x, then the mean A f coincides with the p-th power mean (from now on denoted by P p ); this fact had already been noticed by Knopp [19] before quasi-arithmetic means were formally introduced.
In the course of dealing with the Gaussian iteration process we will use the notation of the Arrow-Pratt index [1,31], which was also investigated by Mikusiński [28]. Whenever f : I → R is twice differentiable with nowhere vanishing first derivative we can define the operator P f := f /f . It can be proved that the comparability of quasi-arithmetic means is equivalent to the pointwise comparability of the respective Arrow-Pratt indexes (see [28] for details).
Following the idea from [30] we will assume that all the considered functions are smooth enough to apply the operator P. Moreover, for technical reasons, we assume that the second derivative is of almost bounded variation (finite variation restricted to every compact interval; cf. [21, p. 135]). Using this definition we introduce the class Obviously, as f = 0, each element belonging to S(I) is a continuous and strictly monotone function, and therefore it generates a quasi-arithmetic mean.
The assumption that f is of almost bounded variation is technical, however important from the point of view of the present paper (this is also the setting which was extensively used in the previous paper [30]).
Following the idea from [30] we are going to deal with the Gaussian iteration of quasi-arithmetic means. Define, for the vector f = (f j ) n j=1 of continuous, strictly monotone functions on I, the mapping A f : I n → I n by In fact A f is the quasi-arithmetic counterpart of the function M, which appears in the definition of invariant mean. Then it is known that there exists a unique continuous and strict mean A ⊗ : I n → I such that A ⊗ • A f = A ⊗ . It also has further implications but let us introduce some necessary notations first. For a vector a of real numbers we denote its arithmetic mean, variance, and spread briefly by a, Var(a), and δ(a) := max(a) − min(a), respectively.
It is known that for every vector a ∈ I n , the sequence (Var(A k f (a))) k∈N tends to zero. Moreover, due to [30], if f ∈ S(I) n then this convergence is double exponential with fractional base. We will prove that, in a non-degenerated case, this sequence tends to zero quadratically and, moreover, we will calculate the limit

Approximate value of quasi-arithmetic means
We are now heading towards the calculation of quasi-arithmetic means in the spirit of Taylor. In fact the crucial identity was already established in the previous paper. Let us recall this result (Riemann-Stieltjes integral is used in its wording).
It was also proved [30,Lemma 4.2] that where the * -norm is defined as g * := sup a, b∈ dom(g) b a g(t)dt . What was not noticed is that if the second derivative of f is locally Lipschitz then the error terms can be majorized much more efficiently. We are going to prove this in a while. For the purpose of this estimation let us make the purely technical assumption K = 1, which will be omitted soon.

Lemma 2.2. For every f ∈ S Lip
1 (I) and a ∈ I n , n ∈ N, where α := 3+7e 3 . Proof. By the mean-value theorem there exist ξ 1 , . . . , ξ n , η ∈ (min a, max a) such that Vol. 92 (2018) On the quasi-arithmetic Gauss-type iteration 1123 We will now prove the second inequality. By (2.1), we have what was to be proved.

Main result
Binding the two results above we can establish the main theorem of the present note. In order to make the notation more compact the brief sum-type notation of means will be used (that is we will write M n k=1 (t k ) instead of M(t 1 , . . . , t n ) ). Additionally, for the same reason, we will use the ± notation of the remainder (with the natural interpretation). where P f : I → R n is defined by P f (x) := (P f1 (x), . . . , P fn (x)), α := 3+7e 3 , and Recall that A and P 2 stand for arithmetic and quadratic means, respectively.
Proof. Applying the machinery described in [30, section 4.1] we can apply the mapping to each function f k . Therefore we will assume, without loss of generality, that K = 1. In fact to make such an assumption possible, we need to verify that the statement in the theorem in both setups are equivalent. Precise calculations are not very simple, but rather straightforward.
In the case when δ(a) ≥ 1 we have C ≥ α 2 e/4 > 1, thus the admissible error on the right hand side is at least 3δ(a) 6 . Meanwhile 6 .
From now on we will assume that δ(a) < 1. By Lemmas 2.1 and 2.2, We know that Var(a) ≤ δ(a) 2 , thus we obtain exp( P f k * )δ(a) 4 .
If we now put a ← A k f (a), we obtain Var(A k+1 f (a)) Var(A k f (a)) 2 But we know that δ(A k f (a)) → 0 and A k f (a) → (A ⊗ (a), . . . , A ⊗ (a)) as k → ∞. Therefore