Generalized Classical Weighted Means, the Invariance, Complementarity and Convergence of Iterates of the Mean-Type Mappings

Under some simple conditions on real function f defined on an interval I, the bivariable functions given by the following formulas Afx,y:=fx+y-fy,Gfx,y:=fxfyy,andHfx,y:=xyfx+y-fy,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{f}\left( x,y\right):= & {} f\left( x\right) +y-f\left( y\right) , \\ G_{f}\left( x,y\right):= & {} \frac{f\left( x\right) }{f\left( y\right) }\,y, \\ \text{ and } \quad H_{f}\left( x,y\right):= & {} \frac{xy}{f\left( x\right) +y-f\left( y\right) }, \end{aligned}$$\end{document}for all x,y∈I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y\in I$$\end{document}, generalize, respectively, the classical weighted arithmetic, geometric and harmonic means. The invariance equations Af∘Gg,Hh=Af,Gg∘Af,Hh=GgandHh∘Af,Gg=Hh,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{f}\circ \left( G_{g},H_{h}\right) =A_{f}, \quad G_{g}\circ \left( A_{f},H_{h}\right) =G_{g} \quad \text{ and } \quad H_{h}\circ \left( A_{f},G_{g}\right) =H_{h}, \end{aligned}$$\end{document}where f, g, h are the unknown functions are, in some special cases, solved. The convergence of iterates of the relevant mean-type mappings is considered. As an application the solutions of some functional equations are determined.


Introduction
The classical Pythagorean harmony proportion involving the bivariable symmetric arithmetic mean A, harmonic mean H and geometric mean G, equivalent to the equality G • (A, H) = G, as well as its extension for the weighted means where t ∈ (0, 1), and referred to as the invariance of the geometric mean with respect to the meantype mappings (A t , H t ), has well known important consequences. In particular it implies that for every t ∈ (0, 1) the sequence ((A t , H t ) n : n ∈ N) of the iterates of the mean-type mapping (A t , H t ) converges to (G, G) (uniformly on compact subsets of (0, ∞) 2 ) [12] (also, under stronger conditions, Borwein and Borwein [2]). This is a special case of the following more general fact. If M, N are continuous bivariable strict means in an interval I, then there is a unique mean K invariant with respect to the mean-type mapping (M, N ), that is satisfying the identity K • (M, N ) = K; moreover the sequence of iterates ((M, N ) n : n ∈ N) converges to (K, K) (uniformly on compact subsets of I 2 ) (see [9,10,12]). At this stage the mean N is called complementary to M with respect to K (briefly, a K-complementary to M ) and vice versa. There is a rich literature related to the invariance equation problems. We refer the interested in the results dealing with invariant means, a survey paper [7]. Let us mention that invariance of the arithmetic mean with respect to the quasi-arithmetic mean-type mappings as well as some related questions were considered among others in [1,[4][5][6]9].
Motivated by these facts, we give necessary and sufficient conditions for a real function f defined on an interval I, under which the functions A f , G f , H f given by the following formulas and the functions f and id| I − f are increasing. Moreover (iii) A f is a weighted arithmetic mean of the weight t ∈ [0, 1], i.e.
Under the assumptions of this remark, the mean A f given by formula (1) is called a generalized weighted arithmetic mean and the function f , called its generator, being Lipschitzian, is absolutely continuous; consequently, it is differentiable almost everywhere.
and the functions f and id| I f are increasing. Moreover (i) G f is a mean if and only if the function f is increasing and x,y∈ I, x < y; Under the assumptions of this remark, the mean G f given by formula (2) is called a generalized weighted geometric mean and the function f , called its generator, is absolutely continuous; consequently, differentiable almost everywhere.
Note the following easy to prove Under the assumptions of this remark, the mean H f given by formula (3) is called a generalized weighted harmonic mean and the function f , called its generator, is absolutely continuous; consequently, is differentiable almost everywhere.
The following result shows that, in particular, the above proposed definitions of generalizations of the classical means are natural. (ii) the geometric mean G is invariant with respect to the mean-type map- to the mean-type mapping (G, G).
Proof. Conditions (i) and (ii) are equivalent (see [9]). Assume (ii). From the definitions of G, A f and H h (see (1) and (3)), equality (4) holds, if and only if, for arbitrary x, y ∈ I, which (after simple calculations) can be written equivalently in the form To prove the "moreover" result note that, in view of Remark 2 (ii), the function f is continuous and, consequently, the mean-type mapping (A f , H f ) is continuous. Since the coordinate means are strict, the result follows from the main result of [10] (see also [12]).

Invariant Means and Some Open Problems
In this section we consider some invariance equations involving the introduced generalized weighted means A f , G f and H f .
We begin with Problem 1. Let I ⊂ (0, ∞) be an interval. Find all functions f, g, h : I → (0, ∞) satisfying the equation assuming that f, g, h are, respectively, the generators of generalized weighted arithmetic, geometric and harmonic means.
In the case when A f is symmetric, i.e. if A f = A, we prove the following then there exist a ∈ (0, ∞) and b ∈ R such that Proof. According to the definitions of the involved means (see (2) and (3)), Eq. (5) reduces to which, after simple calculations, implies that The assumptions of g and h imply that they are absolutely continuous. Let x ∈ I be a differentiability point of g and h. Dividing both sides of the Eq. (6) by x − y, we get It follows that g and h are differentiable at the point x and, letting y → x, gives x, x ∈ I.
Differentiating both sides of (6) with respect to the variable x at the point x, we obtain On the other hand, differentiating both sides of this equality with respect to y (at the points of differentiability of g and h), we get 2y g (x) = (g (y) − g (x)) (h (x) − h (y)) − (g (y) − y g (x)) h (y) and this equality holds true for almost all x and almost all y in I. Setting here y = x we have a.e. in I,  (7), we obtain , a.e. in I, which simplifies to The absolute continuity and strict monotonicity of g imply that g (x) > 0 a.e. in I, so x a.e. in I.
Since the derivative of the absolutely continuous function g coincides a.e. in I with the continuous function x −→ g(x) x , it follows that g must be continuously differentiable in I. Thus g satisfies the differential equation x , x∈ I.

Solving this equation we get
for some a > 0. Hence, in view of (7) we have for some real b. Hence, by the definitions of G g and H h , we get Let us note that Lemma 1 and main results of [10] (also [12]) allow to conclude: We can also consider the invariance of the arithmetic mean A with respect to the mean-type mappings involving at least one of the introduced means A f or H f . Similarly as in the above remark we will get the explicit formulas for the respective mean-type mappings ensuring the invariance, the limit of sequence of its iterates, as well as the complementary means, but we omit statements of these results.
assuming that f, g, h are, respectively, the generators of generalized weighted arithmetic, geometric and harmonic means.
In the case when H h is symmetric, i.e. if H h = H, we prove the following then there exist a ∈ (0, ∞) and b ∈ R such that either Proof. For the same reason as in the previous proof, the functions f and g are differentiable in I. By the definitions of the H, A f and G g (see (1) and (2)), Eq. (8) can be written in the form g(y) y = 2xy x + y , x,y∈ I, x , x,y ∈ I.
Dividing both sides of this equation by x − y we have x,y∈ I, x = y, and letting y tend to x we get Differentiating both sides of (9) in x, (after a simplification) we obtain, for all x, y ∈ I, Now, differentiating both sides of this equality in y, we get, for all x, y ∈ I, Taking here y = x we obtain as, if g (x) − 3xg (x) = 0 then also g (x) − xg (x) = 0, and we would have g (x) = 0, contradicting the assumption. Thus Hence, making use of (10), we get which implies that for every x ∈ I, that is, for every x ∈ I, either g (x) − xg (x) = 0 or g (x) = 0. It implies that either there is a constant a > 0 such that or there is a > 0 such that From (10), in the first case we get f = 0 in I, so there exists b ∈ R such that and in the second case, f = 1 in I, so, for some real b, Consequently, in the first case we get and in the second case, Let us mention here also a result, related to the invariance of the harmonic mean, that gives us the explicit formula for an H-complementary mean to generalized weighted arithmetic mean A f . The result reads as follows Similarly, considering the invariance of the harmonic mean H with respect to the mean-type mappings involving at least one of the introduced means G f or H f one can determine the explicit formulas for the relevant mean-type mappings ensuring the invariance, the limit of sequence of its iterates, as well as the complementary means. Finally we formulate the following