Invariant means, complementary averages of means, and a characterization of the beta-type means

We prove that whenever the selfmapping $(M_1,\dots,M_p)\colon I^p \to I^p$, ($p \in \mathbb{N}$ and $M_i$-s are $p$-variable means on the interval $I$) is invariant with respect to some continuous and strictly monotone mean $K \colon I^p \to I$ then for every nonempty subset $S \subseteq\{1,\dots,p\}$ there exists a uniquely determined mean $K_S \colon I^p \to I$ such that the mean-type mapping $(N_1,\dots,N_p) \colon I^p \to I^p$ is $K$-invariant, where $N_i:=K_S$ for $i \in S$ and $N_i:=M_i$ otherwise. Moreover \begin{equation*} \min(M_i\colon i \in S)\le K_S\le \max(M_i\colon i \in S). \end{equation*} Later we use this result to: (1) construct a broad family of $K$-invariant mean-type mappings, (2) solve functional equations of invariant-type, and (3) characterize Beta-type means.


Introduction
We show that every symmetric and increasing p-variable mean K, that is invariant with respect to a given mean-type mapping M = (M 1 , ..., M p ), generate a unique finite family of mean-type mappings, the coordinates of which are referred to as the complementary K-averages of a respective subfamily of the coordinates means of M. The basic symmetric mean K remains invariant with respect to each member of this family, which, in general does not include any iterates of M. However, according to the invariance principle ( [6], see also, [5] and [8]) the iterates of the each mean-type mapping from this family, converge on compact subsets, to the mean-type map K = (K, ..., K), that is important in effective solving some functional equations (section 4).
In a recent paper [3], a purely structural property of Beta Euler function, gave rise to introduce a family of new means B p , called beta-type means. In this note (section 5) we observe that the beta-type mean can be characterized via invariance identity involving the classical geometric and arithmetic mean.
A function M : I p → I is called a mean in I if The We say that a function K : .. = x p } then there is a unique Minvariant mean K : I p → I and the sequence of iterates (M n ) n∈N of the mean-type mapping M converges to K := (K, . . . , K) pointwise on I p .

A family of complementary means
Recall the following Remark 1. ( [4])Assume that K : I 2 → I is a symmetric mean which is continuous and monotone. Then (i) for an arbitrary mean M 1 : I 2 → I there is a unique mean M 2 : is referred to as a K-complementary mean for M 1 ; and we have In this case p ≥ 3 the counterpart of part (i) of Remark 1 is false which shows the following but a partial counterpart of part (ii) holds true. The main result of this section reads as follows.
Theorem 2. Let M 1 , . . . , M p : I p → I of means. Assume that K : I p → I is a continuous and monotone mean which is invariant with respect to the mean type mapping M := (M 1 , . . . , M p ).
Then for every nonempty subset S ⊆ N p there exists a unique mean K S (M) : Proof. In the case S = N p the K S (M)-invariance of K implies K S (M) = K and the statement is obvious. From now on we assume that S = N p .
Denote briefly M ∨ := max{M i : i ∈ S} and M ∧ := min{M i : i ∈ S}. Fix x ∈ I n arbitrarily. Define a function T : I → I p by and f : I → I by f (α) := K • T (α). Then, as K is continuous and strictly increasing, so is f . Therefore in view of the equality Then we have Then, as f is a monomorphism we obtain The intuition beyond this theorem is the following. Once we have a continuous and monotone mean K such that M is K-invariant mean we can unite a subfamily (M s ) s∈S into a single mean (denoted by K S (M)) to preserve the K-invariance. In view of Theorem 1, such a mean is unique. In this connection we propose the following  (ii) K is monotone, homogeneous and for every S ⊂ N p the iterates of K S (M) converge uniformly on compact subsets to a mean-type mapping K = (K, ..., K); (iii) a function F : (0, ∞) p → R is continuous on the diagonal ∆ ((0, ∞) p ) := {(x 1 , . . . , x p ) ∈ I : x 1 = · · · = x p } and satisfies the functional equation Proof. The homogeneity and monotonicity of M 1 , . . . , M p imply their continuity [6, Theorem 2]), so the invariance principle implies (i). Now we prove that K is monotone. Indeed, take two vectors v, w ∈ (0, ∞) p such that v i ≤ w i for all i ∈ N p and v i0 < w i0 for certain i 0 ∈ N p . Then, as each M i is monotone, there exists a constant θ ∈ (0, 1) such that Then for all n ∈ N and i ∈ N p we have In a limit case as n → ∞ in view of the first part of this statement we obtain K(v) ≤ θK(w) < K(w). Thus K is monotone, which is (ii).
To prove the converse implication, take an arbitrary function ϕ : I → R and put F := ϕ • K. Then, for all x ∈ (0, ∞) p , making use of the K-invariance with respect to M, we have which completes the proof of (iii).
(iv) we omit similar argument.
Part (ii) of this result gives rise to the following 4.1. General complementary process. Once we have a mean-type M : I p → I p and a continuous and monotone mean K : I p → I which is M-invariant let K + (M, K) be the smallest family of mean-type mappings containing M which is closed under K-complementary averaging.
More precisely for every X ∈ K + (M, K) and nonempty subset S ⊆ N p we have K S (X) ∈ K + (M, K), too. We also define a family of means for all X ∈ K 0 (M, K) .
Its inductive proof is obvious in view of Theorem 2 (moreover part). Now we prove that complementary means preserve symmetry.
Proposition 2. If a continuous and monotone mean K : I p → I is invariant with respect to a mean-type mapping M := (M 1 , . . . , M p ) : I p → I p such that all M i -s are symmetric, then K and all means in K 0 (M, K) are symmetric.
Proof. Fix a nonconstant vector x ∈ I p and a permutation σ of N p . First observe that As the family K 0 (M, K) is generating by complementing, we need to show that symmetry is preserved by a single complement. Therefore it is sufficient to show that the mean K S (M) defined in Theorem 2 is symmetric. However, using the notions therein, we have contradicting the above equality. Similarly we exclude the case K S (M)(x) > K S (M)(x • σ). Therefore K S (M)(x) = K S (M)(x • σ) which, as x and σ were taken arbitrarily, yields the symmetry of K S (M).

An applications to Beta-type means
Following [3], for a given k ∈ N we define a p-variable Beta-type mean B p : This is a particular case of so-called biplanar-combinatoric means (Media biplana combinatoria) defined in Gini [1] and Gini-Zappa [2].
In order to formulate the next results, we adapt the notation that A, G and H are arithmetic, geometric and harmonic means of suitable dimension, respectively.
In [7], the invariance G•(A, H) = G, equivalent to the Pythagorean proportion, has been extended for arbitrary p ≥ 3. In case p = 3 it takes the form G• (A, F, H and H ≤ F ≤ A . Hence, making use of Corollary 1 with p = 3, K = G, S = {1, 2} we obtain the following For all x 1 , x 2 , x 3 , the following inequality holds

and the inequalities are sharp for nonconstant vectors
Passing to the main part of this section, first observe the following easy-to-see lemma. We are now going to establish the set K + (B, G). It is quite easy to observe that all means in K 0 (B, G) are of the form H p,α : I p → I (α ∈ R) given by In the next lemma we show some elementary properties of the family (H p,α ).
Lemma 2. Let p ∈ N. Then (i) H p,α is reflexive for all α ∈ R, that is H p,α (x, . . . , x) = x for all x ∈ R + , (ii) H p,α is continuous for all α ∈ R (as a p-variable function), (iii) H p,α is a strict mean for all α ∈ [− 1 p−1 , 1], (iv) H p,α is a symmetric function for all α ∈ R, that is H p,α (x • σ) = H p,α (x) for all x ∈ R p + and a permutation σ of N p , By Theorem 2 there exists exactly one mean G S (H) : I p → I such that G is G S (H)-invariant, where G S (H) : I p → I p is given by On the other hand, in view of Lemma 3 we obtain that G is invariant with respect to the mean-type mapping H 0 : I p → I p given by As G is both G S (H)-invariant and H 0 -invariant we obtain G S (H) = H p,β , and consequently G S (H) = H 0 . Observe that H 0 ∈ Λ is a straightforward implication of the equality i∈Np\S α i + |S|β = i∈Np α i = 0. Now we show that β ∈ Q ∩ [− 1 p−1 , 1], which would complete the proof. But this is simple in view of the definition of β and the analogous property α i ∈ Q∩[− 1 p−1 , 1], which is valid for all i ∈ S.
Remark 3. Let us just put the reader attention that inclusions in the above theorem are strict. More precisely, we can prove by simple induction that the denominator of α in irreducible form has no prime divisors greater than p.