On the Beckenbach–Gini–Lehmer Means and Means Mappings

: Beckenbacg–Gini–Lehmer type means and mean-type mappings generated by functions of several variables, for which the arithmetic mean is invariant, are introduced. Equality of means of that type, their homogeneity, and convergence of the iterates of the respective mean-type mappings are considered. An application to solving a functional equation is given.


Introduction
If I is an interval and ϕ is a positive function defined on I, then the two-variable function M [ϕ] defined on I 2 by M [ϕ] (x, y) = xϕ(x) + yϕ(y) ϕ(x) + ϕ(y) , x, y ∈ I, maps I 2 into I, and even more: it is a mean in I, that is, min(x, y) ≤ M [ϕ] (x, y) ≤ max(x, y) for all x, y ∈ I, and it is called a Gini mean, Beckenbach-Gini mean, or Lehmer mean ( [1][2][3]) of a generator ϕ. In the sequel we shall write briefly, B-G-L mean. The function M * [ϕ] : I 2 → I defined by is also a B-G-L mean in I, because M * and, if ϕ is continuous, the sequence (M [ϕ] , M * [ϕ] ) n : n ∈ N of iterates of the mean-type mapping M [ϕ] , M * [ϕ] converges to the mean-type mapping (A 2 , A 2 ) , uniformly on compact sets ( [4,5]). For this reason M * [ϕ] is called complementary to M [ϕ] (and vice-versa) with respect to the arithmetic mean A 2 ( [6]).

Preliminaries
In the sequel I ⊂ R denotes an interval and k ∈ N, k ≥ 2, a fixed number. A function M : I k → I is said to be a k-variable mean if, for all x 1 , . . . , x k ∈ I, If for all x 1 , . . . , x k ∈ I such that min{x 1 , . . . , x k } < max{x 1 , . . . , x k } (these inequalities are sharp), M is called a strict mean. A mean M is symmetric if for every bijection σ : {1, . . . , k} → {1, . . . , k}.
A function M : I k → I k , M = (M 1 , . . . , M k ), is called a mean-type mapping if, for each i ∈ {1, . . . , k}, the function M i : I k → I is a mean. A mean-type mapping M is called strict if each of its coordinate functions M i is a strict mean.
Let M : I k → I k , M = (M 1 , . . . , M k ), be a mean-type mapping. A mean K : I k → I is called invariant with respect to a mean-type mapping M, shortly, M-invariant if K • M = K; i.e., if, for all x 1 , . . . , x k ∈ I, It turns out that if M : I k → I k , M = (M 1 , . . . , M k ) is a continuous and strict mean-type mapping, then there exists a unique continuous and M-invariant mean K : I k → I. Moreover K is strict and the sequence (M n : n ∈ N) of iterates of the mean-type mapping M converges on I k to the mean-type mapping K = (K, . . . , K) k times (cf. [4,7,8] where a more general result is presented).

Mean-Type Mappings Generated by Functions of a Single Variable
In the sequel, for k ∈ N, k ≥ 2, the symbol A k stands for the arithmetic mean The following extends the result of two-variable B-G-L means in [5]. [ϕ],k : I k → I defined by is a k-variable mean in I (here we adopt the convention, that any sum over the empty set of indices is zero); (ii) The arithmetic mean A k is invariant with respect to the mean-type mapping M [ϕ],k : I k → I k given by i.e., A k • M [ϕ],k we have i.e., M [ϕ],k is a mean. Result (ii) follows immediately from the obvious equality [ϕ],k (x 1 , . . . , x k ) = x 1 + · · · + x k .
Result (iii) is a consequence of (ii) and the main result of [4] (see also [8]).
The mean M [1] [ϕ],k is a natural extension of the two-variable B-G-L mean are complementary with respect to the arithmetic mean A 2 (x, y) = x+y 2 ; i.e., if and only if ϕ = ψ ( [5]). Moreover the M * [ψ],2 is also a B-G-L mean. Note also that x, so M [ϕ],2 (x, y) and M * [ϕ],2 (x, y) are the weighted arithmetic means of x and y with cyclically replaced weights For this reason, the mean-type mapping M [ϕ],k defined by (1) can be called a B-G-L mean-type mapping of a generator ϕ.

Mean-Type Mappings Generated by Functions of Several Variables
In this section we generalize Theorem 1 as follows.
Theorem 2. Let I ⊂ R be an interval and k ∈ N, k ≥ 2, be fixed. Let f : I k−1 → (0, ∞) be an arbitrary function.
[ f ],k ; (iii) If f is continuous then the sequence ,k converges uniformly on compact sets to the mean-type mapping [ f ],k is a mean. We omit similar arguments for the remaining functions. Result (ii) follows from an easy to see equality Result (iii) is a consequence of (ii) and the main result of [4] (see also [8]).
As suggested in the previous section, the mean-type mapping M [ f ],k is referred to as a B-G-L mean-type mapping of the generator f .

Remark 2. The above result remains true on replacing
[ f ],k by each of the following k − 1 mean-type mappings: which arises from M [ f ],k by the cyclic translation of its coordinate means.

Equality M [ f ],k = M [ϕ],k
In this section we examine when the means defined in the previous two sections coincide (and therefore we omit writing upper indexes at M [i] [ f ],k and M [i] [ϕ],k ). We begin with the following if, and only if, for a positive constant c, This remark is an easy consequence of the definitions of the involved means. Its 3-dimensional counterpart reads as follows: Theorem 3. Let f : I 2 → (0, ∞) be a symmetric function and ϕ : I → (0, ∞). Then if, and only if, there is c > 0 such that Proof. Assume that equality (2) x, y ∈ I.
Defining D : we can write this equation in the form Hence, we get The symmetry of f implies that D(y)ϕ(y) which is equivalent to Consequently, there is a constant c > 0 such that x ∈ I.
Since the reversed implication is easy to verify, the proof is completed.
In a similar way, one can prove more general Suppose that a function f : I k−1 → (0, ∞) is symmetric and ϕ : I → (0, ∞). Then if, and only if, there is c > 0 such that Proof. (Proof an idea). Suppose that (4) is satisfied. Putting in (4) by the symmetry of f , we get and, consequently, Returning to the one before last substitution x k−3 = · · · = x 3 = x k−1 , by the symmetry of f and the above, we obtain and finally, repeating this procedure k − 1-times, we come back to (4) getting for all x 1 , ...x k−2 , x k−1 ∈ I. Since f is symmetric, we have which, by (5), gives The reversed implication is obvious.
Hence, taking into account that and the interior of the set where this determinant disappears is empty, the continuity of f and g implies that λ 1 − µ 1 = 0 and λ 2 − µ 2 = 0. Hence and, by the above formulas, x, y, z ∈ I, which shows that the second variable's function [ f ],3 = M [1] [g], 3 . Namely, we have the following.

Remark 6.
Let I ⊂ R be an interval and let f , g : I 2 → (0, ∞) be symmetric functions. Then if, and only if, g = c f for a positive constant c.
Proof. Assume that equality (7) is satisfied. Then, after simple modifications, we get Putting z := x in (8) we obtain x, y ∈ (0, ∞); therefore, by symmetry of f and g, we get Since the left-hand side is symmetric with respect to x and y, and the right-hand side does not depend on y, it follows that and consequently, there is a real constant c > 0 such that which, by virtue of (9), gives the required claim. Since the reversed implication is obvious, the proof is completed.

Homogeneity of the Mean M [ f ],k
In this section we do not consider the mean-type mappings-therefore, similarly to in Section 5, we omit writing upper indexes.
x, t > 0; (ii) If ϕ is measurable or the graph ϕ is not a dense set in (0, ∞) 2 , then there is a p > 0 such that The means M [p],k have the following interesting property.

Invariance of the Arithmetic Mean with Respect to the Mean-Type Mapping with Coordinate B-G-L Means of Different Single Variable Generators
In this section we consider the problem of invariance of the arithmetic mean A k with respect to the B-G-L mean-type mappings (M [ϕ 1 ] , . . . , M [ϕ k ] ).
In the case k = 2 we have the following (cf. [5]).
The next result shows that in the case k ≥ 3 the situation is completely different.

Remark 8.
Note that having proved formula (13) one can finish the proof with a shorter argument. Let us fix an arbitrary x 0 ∈ I. Without any loss of generality we may assume that Hence, setting y = x 0 in (13), we obtain Using this formula to eliminate the function ψ in (13) and then setting x = x 0 gives (ϕ(y) − 1)[2ϕ(y) 2 − 2ϕ(y) + 1] = 0, y ∈ I, and, consequently, ϕ(y) = 1 for all y ∈ I. In the same way one can show that ψ is constant.
An advantage of argument given in the proof is that, with mainly notational difficulties, it can be generalized to the arbitrary k > 3.

An Application
Making use of Theorem 2 we prove the following.
Then the function Φ is invariant with respect to the mean-type mapping M [ f ],k ; that is, Φ satisfies the functional equation [ f ],k , . . . , M [k] [ f ],k ; if and only if there is a continuous single variable function ϕ : I → R such that where A k is the k-variable arithmetic mean.
Proof. Assume first that Φ : Setting ϕ (t) := Φ • (t, . . . , t) , t ∈ I , we conclude that Φ = ϕ • A k . To prove the converse implication, take an arbitrary continuous function ϕ : I → R and put Φ := ϕ • A k . Then Φ is continuous on the diagonal and, for all x = (x 1 , . . . , x k ) ∈ I k , making use of the invariance of A k with respect to the mean-type mapping M [ f ],k , we have which completes the proof.

Remark 9.
The assumption of the continuity of Φ can be omitted if Φ is a mean (see [8]). Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.