Invariant Means, Complementary Averages of Means, and a Characterization of the Beta-Type Means

Abstract

We prove that whenever the selfmapping $(M_1,\dots ,M_p):I^p\to I^p$, ($p\in \mathbb{N}$ and $M_i$-s are p-variable means on the interval I) is invariant with respect to some continuous and strictly monotone mean $K:I^p\to I$ then for every nonempty subset $S\subseteq \{1,\dots ,p\}$ there exists a uniquely determined mean $K_S:I^p\to I$ such that the mean-type mapping $(N_1,\dots ,N_p):I^p\to I^p$ is K-invariant, where $N_i:=K_S$ for $i\in S$ and $N_i:=M_i$ otherwise. Moreover $min(M_i:i\in S)\le K_S\le max(M_i:i\in S).$ 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.

1. Preliminaries

In the whole paper $I\subset \mathbb{R}$ stands for an interval, $p\in \mathbb{N},$$p>1$ is fixed, and ${\mathbb{N}}_p:=\{1,\dots ,p\}$.

A function $M:I^p\to I$ is called a mean on I if

$$min\left(x_1,\dots ,x_p\right)\le M\left(x_1,\dots ,x_p\right)\le max\left(x_1,\dots ,x_p\right),x_1,\dots ,x_p\in I,$$

or, briefly, if

$$min\mathbf{x}\le M\left(\mathbf{x}\right)\le max\mathbf{x},\mathbf{x}=\left(x_1,\dots ,x_p\right)\in I^p.$$

The mean M is called strict, if for all nonconstant vectors $\mathbf{x}$ these inequalities are sharp; and symmetric, if $M\left(x_{\sigma \left(1\right)},\dots ,x_{\sigma \left(p\right)}\right)=M\left(x_1,\dots ,x_p\right)$ for all $x_1,\dots ,x_p\in I$ and all permutations $\sigma$ of the set ${\mathbb{N}}_p$. Mean M is monotone if it is increasing in each of its variables. It is important to emphasize that every strictly monotone mean is strict.

A mapping $\mathbf{M}:I^p\to I^p$ is referred to as mean-type if there exist some means $M_i:I^p\to I$, $i\in {\mathbb{N}}_p$, such that $\mathbf{M}=(M_1,\dots ,M_p)$. For a mean-type mapping $\mathbf{M}:I^p\to I^p$ we denote a projection onto the i-th coordinate by ${\left[\mathbf{M}\right]}_i:I^p\to I$. In this case we obviously have ${\left[\mathbf{M}\right]}_i=M_i$ for all $i\in {\mathbb{N}}_p$.

We say that a function $K:I^p\to \mathbb{R}$ is invariant with respect to $\mathbf{M}$ (briefly $\mathbf{M}$-invariant), if $K\circ \mathbf{M}=K$.

Theorem 1

(Invariance Principle, [1]). If $\mathbf{M}:I^p\to I^p$, $\mathbf{M}=(M_1,\dots ,M_p)$ is a continuous mean-type mapping such that

$$max\mathbf{M}\left(\mathbf{x}\right)-min\mathbf{M}\left(\mathbf{x}\right)<max\left(\mathbf{x}\right)-min\left(\mathbf{x}\right),\mathbf{x}\in I^p\backslash \Delta \left(I^p\right),$$

where $\Delta \left(I^p\right):=\left\{\mathbf{x}=\left(x_1,\dots ,x_p\right)\in I^p:x_1=\dots =x_p\right\}$ then there is a unique $\mathbf{M}$-invariant mean $K:I^p\to I$ and the sequence of iterates ${\left({\mathbf{M}}^n\right)}_{n\in \mathbb{N}}$ of the mean-type mapping $\mathbf{M}$ converges to $\mathbf{K}:=\left(K,\dots ,K\right)$ pointwise on $I^p$.

2. A Family of Complementary Means

Let us recall the following result.

Remark 1

(Matkowski [2]). Assume that $K:I^2\to I$ is a symmetric mean which is continuous and monotone.

Then

(i)

for an arbitrary mean $M_1:I^2\to I$ there is a unique mean $M_2:I^2\to I$ such that K is $(M_1,M_2)$-invariant,

$$K\circ (M_1,M_2)=K;$$

${\mathcal{K}}_{M_1}:=M_2,$ is referred to as a K-complementary mean for $M_1;$ and we have

$${\mathcal{K}}_{M_1}=M_2⟺{\mathcal{K}}_{M_2}=M_1;$$

(ii)

if $M_1,M_2:I^2\to I$ are means such that K is $(M_1,M_2)$-invariant, then there exists a unique mean $M:I^2\to I$ such that

$$min\left(M_1,M_2\right)\le M\le max\left(M_1,M_2\right)$$

and

$$K\circ (M,M)=K;$$

moreover

$$M=K.$$

In this case, $p\ge 3$ the counterpart of part (i) of Remark 1 is false which shows the following

Example 1.

Let $I=\mathbb{R}$, and $K=A$ where $A\left(x_1,x_2,x_3\right)=\frac{x_1+x_2+x_3}{3}$. The functions $M_1\left(x_1,x_2,x_3\right)=\frac{x_1+x_2}{2},$$M_2\left(x_1,x_2,x_3\right)=x_2$ are means in $\mathbb{R}$. Then it is easy to see that there is no mean $M_3$ such that

$$A\circ \left(M_1,M_2,M_3\right)=A,$$

but a partial counterpart of part (ii) holds true.

The main result of this section reads as follows.

Theorem 2.

Let $M_1,\dots ,M_p:I^p\to I$ be means. Assume that $K:I^p\to I$ is a continuous and monotone mean which is invariant with respect to the mean type mapping $\mathbf{M}:=(M_1,\dots ,M_p)$.

Then for every nonempty subset $S\subseteq {\mathbb{N}}_p$ there exists a unique mean $K_S\left(\mathbf{M}\right):I^p\to I$ such that K in ${\mathbf{K}}_S\left(\mathbf{M}\right)$-invariant, where ${\mathbf{K}}_S\left(\mathbf{M}\right):I^p\to I^p$ is given by

$${\left[{\mathbf{K}}_S\left(\mathbf{M}\right)\right]}_i:=\left\{\begin{array}{cc}{K}_S\left(\mathbf{M}\right)\hfill & \mathit{for}i\in S,\hfill \\ M_i\hfill & \mathit{for}i\in {\mathbb{N}}_p\backslash S.\hfill \end{array}\right.$$

(1)

Moreover, $min(M_i:i\in S)\le K_S\left(\mathbf{M}\right)\le max(M_i:i\in S)$.

Proof. 

In the case $S={\mathbb{N}}_p$ the ${\mathbf{K}}_S\left(\mathbf{M}\right)$-invariance of K implies $K_S\left(\mathbf{M}\right)=K$ and the statement is obvious. From now on we assume that $S\ne {\mathbb{N}}_p$.

Denote briefly $M_{\vee }:=max\{M_i:i\in S\}$ and $M_{\wedge }:=min\{M_i:i\in S\}$. Fix $\mathbf{x}\in I^n$ arbitrarily. Define a function $T:I\to I^p$ by

$${\left[T\left(\alpha \right)\right]}_i:=\left\{\begin{array}{cc}{M}_i\left(\mathbf{x}\right)\hfill & i\in {\mathbb{N}}_p\backslash S,\hfill \\ \alpha \hfill & i\in S.\hfill \end{array}\right.$$

and $f:I\to I$ by $f\left(\alpha \right):=K\circ T\left(\alpha \right)$. Then, as K is continuous and strictly increasing, so is f. Therefore in view of the equality $K\circ \mathbf{M}\left(\mathbf{x}\right)=K\left(\mathbf{x}\right)$ we obtain $f\left(M_{\vee }\left(\mathbf{x}\right)\right)\ge K\left(\mathbf{x}\right)$ and $f\left(M_{\wedge }\left(\mathbf{x}\right)\right)\le K\left(\mathbf{x}\right)$. Thus, there exists unique number ${\alpha }_{\mathbf{x}}\in [M_{\wedge }\left(\mathbf{x}\right),M_{\vee }\left(\mathbf{x}\right)]$ such that $f\left({\alpha }_{\mathbf{x}}\right)=K\left(\mathbf{x}\right)$. Now, as $\mathbf{x}\in I^n$ was arbitrary we define $K_S\left(\mathbf{M}\right)\left(\mathbf{x}\right):={\alpha }_{\mathbf{x}}$.

Then we have

$$K\left(\mathbf{x}\right)=f\left({\alpha }_{\mathbf{x}}\right)=f\left(K_S\left(\mathbf{M}\right)\left(\mathbf{x}\right)\right)=\left(K\circ T\circ \left(K_S\left(\mathbf{M}\right)\right)\right)\left(\mathbf{x}\right)=\left(K\circ \left({\mathbf{K}}_S\left(\mathbf{M}\right)\right)\right)\left(\mathbf{x}\right),$$

which shows that K is ${\mathbf{K}}_S\left(\mathbf{M}\right)$-invariant.

Now we need to show that $K_S\left(\mathbf{M}\right)$ is uniquely determined. Assume that K is ${\mathbf{K}}_S\left(\mathbf{M}\right)$-invariant and $K_S\left(\mathbf{M}\right)\left(\mathbf{x}\right)\ne {\alpha }_{\mathbf{x}}$ for some $\mathbf{x}\in I^p$. Then, as f is a monomorphism we obtain

$$K\circ \left({\mathbf{K}}_S\left(\mathbf{M}\right)\right)\left(\mathbf{x}\right)=\left(K\circ T\circ \left(K_S\left(\mathbf{M}\right)\right)\right)\left(\mathbf{x}\right)\ne f\left({\alpha }_{\mathbf{x}}\right)=K\left(\mathbf{x}\right)$$

contradicting the ${\mathbf{K}}_S\left(\mathbf{M}\right)$-invariance. □

Let us underline that we do not assume that means $M_1,\dots ,M_P$ are continuous. This is relatively recent approach to invariant property which was studied by the authors in [3].

The intuition beyond this theorem is the following. Once we have a continuous and monotone mean K such that $\mathbf{M}$ is K-invariant mean we can unite a subfamily ${\left(M_s\right)}_{s\in S}$ into a single mean (denoted by $K_S\left(\mathbf{M}\right)$) to preserve the K-invariance. In view of Theorem 1, such a mean is unique. In this connection we propose the following

Definition 1.

Let $K:I^p\to I$ be a continuous and monotone mean which is invariant with respect to the mean type mapping $\mathbf{M}:=(M_1,\dots ,M_p)$.

For each set $S\subset {\mathbb{N}}_p:$

(i) 

the mean $K_S\left(\mathbf{M}\right)$ is called a K-complementary averaging of the means $\{M_i:i\in S\}$ with respect to the invariant mean-type mapping $\mathbf{M}=(M_1,\dots ,M_p)$;

(ii) 

the mean-type mapping ${\mathbf{K}}_S\left(\mathbf{M}\right)$ given by (1) is called a K-complementary averaging of the mean-type mapping $\mathbf{M}=(M_1,\dots ,M_p)$ with respect to the means $\{M_i:i\in S\}$.

Moreover, the set

$$\mathfrak{K}\left(K,\mathbf{M}\right):=\left\{{\mathbf{K}}_S\left(\mathbf{M}\right):S\subseteq {\mathbb{N}}_p,S\ne \varnothing \right\}$$

is called the family of all K-complementary averaging of the mean-type mapping $\mathbf{M}=(M_1,\dots ,M_p).$

We can now reapply this result to the complementary of the establish a K-complementary of ${\mathbf{K}}_S\left(\mathbf{M}\right)$ for the set ${\mathbb{N}}_p\backslash S$. More precisely we obtain

Corollary 1.

Under the assumptions of Theorem 2 there exists unique mean $K_S^{*}\left(\mathbf{M}\right):I^p\to I$ such that K is ${\mathbf{K}}_S^{*}\left(\mathbf{M}\right)$-invariant, where ${\mathbf{K}}_S^{*}\left(\mathbf{M}\right):I^p\to I^p$ is given by

$${\left[{\mathbf{K}}_S^{*}\left(\mathbf{M}\right)\right]}_i:=\left\{\begin{array}{cc}{K}_S\left(\mathbf{M}\right)\hfill & \mathit{for}i\in S,\hfill \\ K_S^{*}\left(\mathbf{M}\right)\hfill & \mathit{for}i\in {\mathbb{N}}_p\backslash S.\hfill \end{array}\right.$$

Moreover, $min(M_i:i\in {\mathbb{N}}_p\backslash S)\le K_S^{*}\left(\mathbf{M}\right)\le max(M_i:i\in {\mathbb{N}}_p\backslash S)$.

Let us underline that the value $K_S^{*}\left(\mathbf{M}\right)$ does not depend on $\mathbf{M}$ explicitly. The whole system of dependences is illustrated in Figure 1.

Observe that, as the mean $K_S^{*}\left(\mathbf{M}\right)$ is uniquely determined, we obtain

$${\mathbf{K}}_S^{*}\left(\mathbf{M}\right)\in \mathfrak{K}\left(K,\mathbf{M}\right)\iff \mathrm{all}\mathrm{means}(M_i:i\in {\mathbb{N}}_p\backslash S)\mathrm{are}\mathrm{equal}\mathrm{to}\mathrm{each}\mathrm{other}.$$

3. Application in Solving Functional Equations

Theorem 3.

Let $\mathbf{M}=(M_1,\dots ,M_p)$ be a mean-type mapping such that $M_1,\dots ,M_p:{\left(0,\infty \right)}^p\to \left(0,\infty \right)$ are strictly monotonic and homogeneous. Then

(i) 

the sequence $\left({\mathbf{M}}^n:n\in \mathbb{N}\right)$ of iterates of $\mathbf{M}$ converge uniformly on compact subsets to a mean-type mapping $\mathbf{K}=\left(K,\dots ,K\right)$, where K is a unique $\mathbf{M}$-invariant mean.

(ii) 

K is monotone, homogeneous and for every $S\subset {\mathbb{N}}_p$ the iterates of ${\mathbf{K}}_S\left(\mathbf{M}\right)$ converge uniformly on compact subsets to a mean-type mapping $\mathbf{K}=\left(K,\dots ,K\right)$;

(iii) 

a function $F:{\left(0,\infty \right)}^p\to \mathbb{R}$ is continuous on the diagonal $\Delta \left({\left(0,\infty \right)}^p\right):=\left\{\left(x_1,\dots ,x_p\right)\in {\left(0,\infty \right)}^p:x_1=\dots =x_p\right\}$ and satisfies the functional equation

$$F\circ \mathbf{M}=F$$

(2)

if and only if $F=\phi \circ K$, where $\phi :\left(0,\infty \right)\to \mathbb{R}$ is an arbitrary continuous function of a single variable;

(iv) 

a function $F:{\left(0,\infty \right)}^p\to \mathbb{R}$ is continuous on the diagonal $\Delta \left({\left(0,\infty \right)}^p\right)$ and satisfies the simultaneous system of functional equations

$$F\circ {\mathbf{K}}_S\left(\mathbf{M}\right)=F,S\subset {\mathbb{N}}_p;$$

(3)

if and only if $F=\phi \circ K$, where $\phi :\left(0,\infty \right)\to \mathbb{R}$ is an arbitrary continuous function of a single variable (so (2) and (3) are equivalent)

Proof. 

The homogeneity and monotonicity of $M_1,\dots ,M_p$ imply their continuity (cf. ([4], Theorem 2)), so the invariance principle implies (i).

Now we prove that K is monotone. Indeed, take two vectors $v,w\in {(0,\infty )}^p$ such that $v_i\le w_i$ for all $i\in {\mathbb{N}}_p$ and $v_{i_0}<w_{i_0}$ for certain $i_0\in {\mathbb{N}}_p$. Then, as each $M_i$ is monotone, there exists a constant $\theta \in (0,1)$ such that $M_i\left(v\right)\le \theta M_i\left(w\right)$ for all $i\in {\mathbb{N}}_p$.

Then for all $n\in \mathbb{N}$ and $i\in {\mathbb{N}}_p$ we have

$$\begin{array}{cc}\hfill {\left[{\mathbf{M}}^n\left(v\right)\right]}_i& ={\left[{\mathbf{M}}^{n-1}(M_1\left(v\right),\dots ,M_p\left(v\right))\right]}_i\le {\left[{\mathbf{M}}^{n-1}(\theta M_1\left(w\right),\dots ,\theta M_p\left(w\right))\right]}_i\hfill \\ \hfill & =\theta {\left[{\mathbf{M}}^{n-1}(M_1\left(w\right),\dots ,M_p\left(w\right))\right]}_i=\theta {\left[{\mathbf{M}}^n\left(w\right)\right]}_i.\hfill \end{array}$$

In a limit case as $n\to \infty$ in view of the first part of this statement we obtain $K\left(v\right)\le \theta K\left(w\right)<K\left(w\right)$. Thus, K is monotone, which is (ii).

(iii) Assume first that $F:{\left(0,\infty \right)}^p\to \mathbb{R}$ that is continuous on the diagonal $\Delta \left({\left(0,\infty \right)}^p\right)$ and satisfies Equation (2). Hence, by induction,

$$F=F\circ {\mathbf{M}}^n,n\in \mathbb{N},$$

By (ii) the sequence $\left({\mathbf{M}}^n:n\in \mathbb{N}\right)$ converges to $\mathbf{K}=\left(K,\dots ,K\right)$. Since F is continuous on $\Delta \left({\left(0,\infty \right)}^p\right)$, we hence get for all $\mathbf{x}\in {\left(0,\infty \right)}^p$,

$$\begin{array}{ccc}\hfill F\left(\mathbf{x}\right)& =& \underset{n\to \infty }{lim}F\left({\mathbf{M}}^n\left(\mathbf{x}\right)\right)=F\left(\underset{n\to \infty }{lim}{\mathbf{M}}^n\left(\mathbf{x}\right)\right)=F\left(\mathbf{K}\left(\mathbf{x}\right)\right)\hfill \\ & =& F\left(K\left(\mathbf{x}\right),\dots ,K\left(\mathbf{x}\right)\right).\hfill \end{array}$$

Setting

$$\phi \left(t\right):=F\left(t,\dots ,t\right),t\in \left(0,\infty \right),$$

we hence get $F\left(\mathbf{x}\right)=\phi \left(K\left(\mathbf{x}\right)\right)$ for all $\mathbf{x}\in {\left(0,\infty \right)}^p$.

To prove the converse implication, take an arbitrary function $\phi :I\to \mathbb{R}$ and put $F:=\phi \circ K$. Then, for all $\mathbf{x}\in {\left(0,\infty \right)}^p,$ making use of the K-invariance with respect to $\mathbf{M}$, we have

$$F\left(\mathbf{M}\left(\mathbf{x}\right)\right)=\left(\phi \circ K\right)\left(\mathbf{M}\left(\mathbf{x}\right)\right)=\phi \left(K\left(\mathbf{M}\left(\mathbf{x}\right)\right)\right)=\phi \left(K\left(\mathbf{x}\right)\right)=F\left(\mathbf{x}\right),$$

which completes the proof of (iii).

(iv) we omit similar argument. □

Part (ii) of this result gives rise to the following extension.

General Complementary Process

Once we have a mean-type $\mathbf{M}:I^p\to I^p$ and a continuous and monotone mean $K:I^p\to I$ which is $\mathbf{M}$-invariant let ${\mathfrak{K}}^{+}\left(\mathbf{M},K\right)$ be the smallest family of mean-type mappings containing $\mathbf{M}$ which is closed under K-complementary averaging.

More precisely, for every $\mathbf{X}\in {\mathfrak{K}}^{+}\left(\mathbf{M},K\right)$ and nonempty subset $S\subseteq {\mathbb{N}}_p$ we have ${\mathbf{K}}_S\left(\mathbf{X}\right)\in {\mathfrak{K}}^{+}\left(\mathbf{M},K\right)$, too. We also define a family of means

$${\mathfrak{K}}_0\left(\mathbf{M},K\right):=\left\{{\left[\mathbf{X}\right]}_i:\mathbf{X}\in {\mathfrak{K}}^{+}\left(\mathbf{M},K\right)\mathrm{and}i\in {\mathbb{N}}_p\right\}$$

Obviously using notions from Theorem 2 and Corollary 1 we have

$${\mathfrak{K}}^{+}\left(\mathbf{M},K\right)\supseteq {\mathfrak{K}}^{+}\left({\mathbf{K}}_S\left(\mathbf{M}\right),K\right)\supseteq {\mathfrak{K}}^{+}\left({\mathbf{K}}_S^{*}\left(\mathbf{M}\right),K\right)$$

Furthermore, we have the following.

Proposition 1.

Given an interval $I\subset \mathbb{R}$, $p\in \mathbb{N}$, and a mean-type mapping $\mathbf{M}:=(M_1,\dots ,M_p):I^p\to I^p$ which is invariant with respect to some continuous and monotone mean $K:I^p\to I$. Then

$$min(M_1,\dots ,M_p)\le X\le max(M_1,\dots ,M_p)\mathit{for}\mathit{all}X\in {\mathfrak{K}}_0\left(\mathbf{M},K\right).$$

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 $K:I^p\to I$ is invariant with respect to a mean-type mapping $\mathbf{M}:=(M_1,\dots ,M_p):I^p\to I^p$ such that all $M_i$-s are symmetric, then K and all means in ${\mathfrak{K}}_0\left(\mathbf{M},K\right)$ are symmetric.

Proof. 

Fix a nonconstant vector $x\in I^p$ and a permutation $\sigma$ of ${\mathbb{N}}_p$. First observe that $K\left(x\right)=K\circ \mathbf{M}\left(x\right)=K\circ \mathbf{M}(x\circ \sigma )=K(x\circ \sigma )$, which implies that K is symmetric.

As the family ${\mathfrak{K}}_0\left(\mathbf{M},K\right)$ 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 $K_S\left(\mathbf{M}\right)$ defined in Theorem 2 is symmetric. However, using the notions therein, we have

$$K\circ \left({\mathbf{K}}_S\left(\mathbf{M}\right)\right)\left(x\right)=K\left(x\right)=K(x\circ \sigma )=K\circ \left({\mathbf{K}}_S\left(\mathbf{M}\right)\right)(x\circ \sigma ).$$

By monotonicity of K, if $K_S\left(\mathbf{M}\right)\left(x\right)<K_S\left(\mathbf{M}\right)(x\circ \sigma )$ we would have

$$K\circ \left({\mathbf{K}}_S\left(\mathbf{M}\right)\right)\left(x\right)=K\left(x\right)<K\circ \left({\mathbf{K}}_S\left(\mathbf{M}\right)\right)(x\circ \sigma )$$

contradicting the above equality. Similarly we exclude the case $K_S\left(\mathbf{M}\right)\left(x\right)>K_S\left(\mathbf{M}\right)(x\circ \sigma )$. Therefore $K_S\left(\mathbf{M}\right)\left(x\right)=K_S\left(\mathbf{M}\right)(x\circ \sigma )$ which, as x and $\sigma$ were taken arbitrarily, yields the symmetry of $K_S\left(\mathbf{M}\right)$. □

4. An Applications to Beta-Type Means

Following [5], for a given $k\in \mathbb{N}$ we define a p-variable Beta-type mean ${\mathcal{B}}_p:{\mathbb{R}}_{+}^k\to {\mathbb{R}}_{+}$ by

$${\mathcal{B}}_p(x_1,\dots ,x_p):={\left(\frac{p{x}_1\dots x_p}{x_1+\dots +x_p}\right)}^{\frac{1}{p-1}}.$$

This is a particular case of so-called biplanar-combinatoric means (Media biplana combinatoria) defined in Gini [6] and Gini–Zappa [7].

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 [8], the invariance $G\circ \left(A,H\right)=G,$ equivalent to the Pythagorean proportion, has been extended for arbitrary $p\ge 3$. In case $p=3$ it takes the form $G\circ \left(A,F,H\right)=G$, where

$$F\left(x_1,x_2,x_3\right)=:\frac{x_2{x}_3+x_3{x}_1+x_1{x}_2}{x_1+x_2+x_3},x_1,x_2,x_3>0,$$

and $H\le F\le A$. Hence, making use of Corollary 1 with $p=3,$$K=G$, $S=\left\{1,2\right\}$ we obtain the following.

Remark 2.

For all $x_1,x_2,x_3,$ the following inequality holds

$$H\left(x_1,x_2,x_3\right)\le {\mathcal{B}}_3(x_1,x_2,x_p)\le A\left(x_1,x_2,x_3\right),$$

and the inequalities are sharp for nonconstant vectors $x=\left(x_1,x_2,x_3\right)\in {\left(0,\infty \right)}^3$.

Passing to the main part of this section, first observe the following easy-to-see lemma.

Lemma 1.

Let $p\in \mathbb{N}$, $p\ge 2$. Then there exists exactly one mean $M:I^p\to I$ such that $G\circ \left(A,\underset{(p-1)\mathrm{times}}{\underbrace{M,\dots ,M}}\right)=G$. Furthermore $M={\mathcal{B}}_p$.

Its simple proof which is a straightforward implication of Theorem 2 is omitted. Based on this lemma it is natural to define a mean-type mapping $\mathbf{B}:I^p\to I^p$ by ${\left[\mathbf{B}\right]}_1:=A$ and ${\left[\mathbf{B}\right]}_i:={\mathcal{B}}_p$ for all $i\in \{2,\dots ,p\}$. Then we have $G\circ \mathbf{B}=G$, which implies that the geometric mean is the unique $\mathbf{B}$-invariant mean.

We are now going to establish the set ${\mathfrak{K}}^{+}\left(\mathbf{B},G\right)$. It is quite easy to observe that all means in ${\mathfrak{K}}_0\left(\mathbf{B},G\right)$ are of the form ${\mathcal{H}}_{p,\alpha }:I^p\to I$ ($\alpha \in \mathbb{R}$) given by

$${\mathcal{H}}_{p,\alpha }(x_1,\dots ,x_p):={\left(x_1\dots x_p\right)}^{\frac{1-\alpha }{p}}{\left(\frac{x_1+\dots +x_p}{p}\right)}^{\alpha }$$

including ${\mathcal{B}}_p={\mathcal{H}}_{p,-\frac{1}{p-1}}$. In the next lemma we show some elementary properties of the family $\left({\mathcal{H}}_{p,\alpha }\right)$.

Lemma 2.

Let $p\in \mathbb{N}$. Then

1.

${\mathcal{H}}_{p,\alpha }$ is reflexive for all $\alpha \in \mathbb{R}$, that is ${\mathcal{H}}_{p,\alpha }(x,\dots ,x)=x$ for all $x\in {\mathbb{R}}_{+}$,

2.

${\mathcal{H}}_{p,\alpha }$ is continuous for all $\alpha \in \mathbb{R}$ (as a p-variable function),

3.

${\mathcal{H}}_{p,\alpha }$ is a strict mean for all $\alpha \in [-\frac{1}{p-1},1]$,

4.

${\mathcal{H}}_{p,\alpha }$ is a symmetric function for all $\alpha \in \mathbb{R}$, that is ${\mathcal{H}}_{p,\alpha }(x\circ \sigma )={\mathcal{H}}_{p,\alpha }\left(x\right)$ for all $x\in {\mathbb{R}}_{+}^p$ and a permutation σ of ${\mathbb{N}}_p$,

5.

${\mathcal{H}}_{p,1}$ and ${\mathcal{H}}_{p,0}$ are p-variable arithmetic and geometric means, respectively,

6.

${\mathcal{H}}_{p,\alpha }$ is increasing with respect to α, that is ${\mathcal{H}}_{p,\alpha }\left(x\right)<{\mathcal{H}}_{p,\beta }\left(x\right)$ for every nonconstant vector $x\in {\mathbb{R}}_{+}^p$ and $\alpha ,\beta \in \mathbb{R}$ with $\alpha <\beta$.

Proof. 

By the definition of ${\mathcal{H}}_{p,\alpha }$ we can easily verify that (1), (2), (4) and (5) holds.

From now on fix a nonconstant vector $x=(x_1,\dots ,x_p)\in {\mathbb{R}}_{+}^p$. By (4) we may assume without loss of generality that $x_1\le \dots \le x_p$. Denote briefly

$$g:=\sqrt[p]{x_1\dots x_p}\mathrm{and}a:=\frac{x_1+\dots +x_p}{p}.$$

By Cauchy inequality we have $x_1<g<a<x_p$. Moreover, by the definition ${\mathcal{H}}_{p,\alpha }\left(x\right)=a^{\alpha }{g}^{1-\alpha }$. Thus, for all $\alpha <\beta$ we have

$${\mathcal{H}}_{p,\beta }\left(x\right)=a^{\beta }{g}^{1-\beta }=a^{\alpha }{g}^{1-\alpha }{\left(\frac{a}{g}\right)}^{\beta -\alpha }>a^{\alpha }{g}^{1-\alpha }={\mathcal{H}}_{p,\alpha }\left(x\right),$$

which completes the proof of (6). The only remaining part to be proved is (3). However, applying (6), it is sufficient to show that

$${\mathcal{H}}_{p,1}\left(x\right)<max\left(x\right)=x_p\mathrm{and}{\mathcal{H}}_{p,-\frac{1}{p-1}}\left(x\right)>min\left(x\right)=x_1.$$

By (5) we immediately obtain ${\mathcal{H}}_{p,1}\left(x\right)=a<x_p$. For the second part observe that

$$\begin{array}{c}\hfill {\mathcal{H}}_{p,-\frac{1}{p-1}}\left(x\right)=\sqrt[p-1]{\frac{g^p}{a}}=\sqrt[p-1]{\frac{x_1\dots x_p}{a}}>\sqrt[p-1]{\frac{x_1\dots x_p}{x_p}}=\sqrt[p-1]{x_1\dots x_{p-1}}\ge x_1,\end{array}$$

which completes the proof. □

Now we generalize Lemma 1 to the following form

Lemma 3.

Let $p\in \mathbb{N}$, $p\ge 2$ and $\alpha \in {\mathbb{R}}^p$. Then $G\circ ({\mathcal{H}}_{p,{\alpha }_1},\dots ,{\mathcal{H}}_{p,{\alpha }_p})=G$, if and only if ${\alpha }_1+\dots +{\alpha }_p=0$.

Its proof is obvious in view of the identity $G\circ ({\mathcal{H}}_{p,{\alpha }_1},\dots ,{\mathcal{H}}_{p,{\alpha }_p})={\mathcal{H}}_{p,\frac{1}{p}({\alpha }_1+\dots +{\alpha }_p)}$. Having this proved, let us show the next important result.

Theorem 4.

Let $p\ge 2$. Then

$$\begin{array}{cc}\hfill & {\mathfrak{K}}^{+}\left((A,\underset{(p-1)\mathrm{times}}{\underbrace{{\mathcal{B}}_p,\dots ,{\mathcal{B}}_p}}),G\right)\hfill \\ \hfill & \subseteq \left\{({\mathcal{H}}_{p,{\alpha }_1},\dots ,{\mathcal{H}}_{p,{\alpha }_p})\mid {\alpha }_1,\dots ,{\alpha }_p\in \mathbb{Q}\cap [-\frac{1}{p-1},1]\mathit{and}{\alpha }_1+\dots +{\alpha }_p=0\right\}.\hfill \end{array}$$

In particular

$${\mathfrak{K}}_0=\left((A,{\mathcal{B}}_p,\dots ,{\mathcal{B}}_p),G\right)\subseteq \left\{{\mathcal{H}}_{p,\alpha }\mid \alpha \in \mathbb{Q}\cap [-\frac{1}{p-1},1]\right\}.$$

Proof. 

First observe that in view of Lemma 1 G is $\mathbf{B}$-invariant, and thus the set ${\mathfrak{K}}^{+}(\mathbf{B},G)$ is well-defined. Now denote briefly

$$\mathsf{\Lambda }:=\left\{({\mathcal{H}}_{p,{\alpha }_1},\dots ,{\mathcal{H}}_{p,{\alpha }_p})\mid {\alpha }_1,\dots ,{\alpha }_p\in \mathbb{Q}\cap [-\frac{1}{p-1},1]\mathrm{and}{\alpha }_1+\dots +{\alpha }_p=0\right\}.$$

Obviously $\mathbf{B}\in \mathsf{\Lambda }$, so it is sufficient to prove that $\mathsf{\Lambda }$ is closed with respect to G-complementary averaging. To this end take an arbitrary vector $({\alpha }_1,\dots ,{\alpha }_p)$ real numbers such that $\mathbf{H}:=({\mathcal{H}}_{p,{\alpha }_1},\dots ,{\mathcal{H}}_{p,{\alpha }_p})\in \mathsf{\Lambda }$ and a nonempty subset $S\subset {\mathbb{N}}_p$.

By Theorem 2 there exists exactly one mean $G_S\left(\mathbf{H}\right):I^p\to I$ such that G is ${\mathbf{G}}_S\left(\mathbf{H}\right)$-invariant, where ${\mathbf{G}}_S\left(\mathbf{H}\right):I^p\to I^p$ is given by

$${\left[{\mathbf{G}}_S\left(\mathbf{H}\right)\right]}_i:=\left\{\begin{array}{cc}{\mathcal{H}}_{p,{\alpha }_i}\hfill & \mathrm{for}i\in {\mathbb{N}}_p\backslash S,\hfill \\ G_S\left(\mathbf{H}\right)\hfill & \mathrm{for}i\in S.\hfill \end{array}\right.$$

On the other hand, in view of Lemma 3 we obtain that G is invariant with respect to the mean-type mapping ${\mathbf{H}}_0:I^p\to I^p$ given by

$${\left[{\mathbf{H}}_0\right]}_i:=\left\{\begin{array}{cc}{\mathcal{H}}_{p,{\alpha }_i}\hfill & \mathrm{for}i\in {\mathbb{N}}_p\backslash S,\hfill \\ {\mathcal{H}}_{p,\beta }\hfill & \mathrm{for}i\in S,\hfill \end{array}\right.\mathrm{where}\beta =\frac{1}{\left|S\right|}\sum _{i\in S}{\alpha }_i.$$

As G is both ${\mathbf{G}}_S\left(\mathbf{H}\right)$-invariant and ${\mathbf{H}}_0$-invariant we obtain $G_S\left(\mathbf{H}\right)={\mathcal{H}}_{p,\beta }$, and consequently ${\mathbf{G}}_S\left(\mathbf{H}\right)={\mathbf{H}}_0$. Observe that ${\mathbf{H}}_0\in \mathsf{\Lambda }$ is a straightforward implication of the equality

$$\sum _{i\in {\mathbb{N}}_p\backslash S}{\alpha }_i+\left|S\right|\beta =\sum _{i\in {\mathbb{N}}_p}{\alpha }_i=0.$$

Now we show that $\beta \in \mathbb{Q}\cap [-\frac{1}{p-1},1]$, which would complete the proof. However, this is easy in view of the definition of $\beta$ and the analogous property ${\alpha }_i\in \mathbb{Q}\cap [-\frac{1}{p-1},1]$, which is valid for all $i\in S$. □

Remark 3.

Let us emphasize 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.