Quasi-Cauchy quotients and means

Let I⊂R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\subset {\mathbb {R}}$$\end{document} be an interval that is closed under addition, and k∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k\in {\mathbb {N}}$$\end{document}, k≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2\,$$\end{document}. For a function f:I→0,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:I\rightarrow \left( 0,\infty \right) $$\end{document} such that Fx:=fkxkfx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\left( x\right) :=\frac{f\left( kx\right) }{ kf\left( x\right) }$$\end{document} is invertible in I, the k-variable function Mf:Ik→I,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M_{f}:I^{k}\rightarrow I,$$\end{document}Mfx1,…,xk:=F-1fx1+⋯+xkfx1+⋯+fxk,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} M_{f}\left( x_{1},\ldots ,x_{k}\right) :=F^{-1}\left( \frac{f\left( x_{1}+\cdots +x_{k}\right) }{f\left( x_{1}\right) +\cdots +f\left( x_{k}\right) } \right) , \end{aligned}$$\end{document}is a premean in I, and it is referred to as a quasi Cauchy quotient of the additive type of generator f. Three classes of means of this type generated by the exponential, logarithmic, and power functions, are examined. The suitable quasi Cauchy quotients of the exponential types (for continuous additive, logarithmic, and power functions) are considered. When I is closed under multiplication, the quasi Cauchy quotient means of logarithmic and multiplicative type are studied. The equalities of premeans within each of these classes are discussed and some open problems are proposed.


Introduction
Let I be a real interval that is closed under addition, and k ≥ 2 be a fixed positive integer. For a function f : I → (0, ∞) , the k-variable function called a Cauchy quotient of the additive type (generated by f ), arises when we divide the left-hand side by the right-hand side of the Cauchy functional equation for the additive function f (x 1 + · · · + x k ) = f (x 1 ) + · · · + f (x k ) , written in the customary way. The restriction of this function to the main diagonal of I k leads to the single variable function F (x) := f (kx) kf (x) . Theorem 1 in Sect. 3 gives general conditions under which F is invertible, and the kvariable function M f : I k → R, is a premean or a mean, which is referred to as a quasi-Cauchy quotient of additive-type premean or mean. Of course, additive functions of the form f (x) = px are useless here. But it is natural to ask what kind of means one can get if the generating function f belongs to one of the three remaining elementary classes of functions: exponential, logarithmic and multiplicative, characterized by the respective Cauchy functional equation. Answering this question we show that the one-parameter family of exponential functions f (x) = p x generates a new class of means (Proposition 1); the logarithmic functions log p lead to a single premean that is not a mean in any subinterval (Proposition 2); and in the case of power functions, the family of premeans is empty (Remark 8). In the context of Proposition 1, a mean that is invariant with respect to some mean-type mapping appears (Corollary 1).
Since F is invertible iff so is the function 1 F , and there is no need to consider the quasi-Cauchy quotients separately if the roles of nominator and denominator are reversed (Sect. 2, Remarks 3 and 4, concerning also the remaining sections). In Sect. 4, Theorem 2 gives conditions under which the quasi-Cauchy quotient of exponential-type, i.e.the function ] k , is a premean or a mean. For the continuous additive functions f (x) = px (p > 0) one gets a k-variable mean (independent of p) which in the case k = 2 coincides with the harmonic mean (Proposition 3). These means, called Beta-type means, B k : (0, ∞) k → (0, ∞) , i.e. f (x) = x p for some p ∈ R, p = 0, then, independently of p, the quasi-Cauchy quotient mean generated by f is the Beta-type mean (Proposition 5). Let I be a real interval that is closed under addition. In Sect. 5, Theorem 3 provides conditions under which the quasi-Cauchy quotient of logarithmic type , is a premean or a mean. In this case the continuous additive functions f (x) = px (p = 0) generate Beta-type means (Proposition 6), obtained also in the previous section. The exponential functions produce a new class of means described for arbitrary k ≥ 3 by an implicit equation, and effectively for k = 2 (Proposition 7). It is interesting that power functions generate an extended family of Beta-type means (Proposition 8).
In Sect. 6, Theorem 4 gives conditions guaranteeing that the quasi-Cauchy quotient of multiplicative type is a premean or a mean. Applying Theorem 4 for the exponential functions f (x) = p x one gets a new family of means given, in general, by an implicit formula, which in case k = 2 reduces to the (translated) geometric mean M f (x, y) = 1 + (x − 1) (y − 1) for all x, y > 1 (Proposition 9). Proposition 10 describes the means generated by Theorem 4 and the logarithmic functions. Moreover it turns out that in this case the additive continuous functions do not produce any means (Remark 18). The Beta-type mean B k plays an important role here, as it appears naturally in each of the sections and helps to describe some of the obtained classes of means.
After each theorem, we pose a problem concerning the equality of the introduced mean. We give only a partial solution of Problem 4, concerning the equality of two multiplicative-type quasi-Cauchy premeans. The suitable functional equations restricted to the diagonal are related to the problem posed by Reich [9].
Let us note that the question when the quasi-Cauchy difference is a mean is considered in [8].

Preliminaries
Let I ⊂ R be an interval and k ∈ N, k ≥ 2, be fixed. Recall that a function M : A k-variable mean or premean in I is called: Of course, every mean is a premean, but the converse implication is not true. However we have the following Remark 1. If a function M : I k → R is reflexive and (strictly) increasing in each variable, then it is a (strict) mean; in the sequel it is referred to as a (strictly) increasing k-variable mean.
In this connection we have

Remark 2.
There is no mean that is decreasing with respect to any variable.
Note the following general method of construction of premeans and means.

Lemma 1.
Let k ∈ N, k ≥ 2, I ⊂ R be an interval. For a function g : Then: (i) if g I k ⊂ γ (I) and γ is invertible, then the function Remark 4. For a closed under addition interval I, a continuous and strictly monotonic function f : I → (0, ∞) , and k ∈ N, let (the left-hand side and the right-hand side of the additivity equation By the previous remark, if the function l r is invertible, then, Similar equalities hold true for the exponential-type, logarithmic type, and multiplicative-(or power)-type Cauchy quotients.

Additive quasi-Cauchy quotient
Theorem 1. Let I ⊂ R be an interval that is closed under addition, i.e. I + I ⊂ I; f : I → (0, ∞) be a function; k ∈ N, k ≥ 2; and let F : I → (0, ∞) be given by Then (i) if F is one-to-one and then the function M f : is a correctly defined symmetric k-variable premean in I; (ii) if f is continuous and the function is strictly increasing (decreasing) in one of the variables, then F is strictly increasing (decreasing) and the function M f defined by (3) is a continuous symmetric strictly increasing k-variable mean in I. (1) and (3), for every x ∈ I we have To show (ii) assume that the function (4) is strictly increasing in one of the variables. The symmetry of this function implies that it is strictly increasing in each of the variables. Hence, if x, y ∈ I, x < y, then The function M f , being the composition of the function (4) and F −1 , is (strictly) increasing.
Since, in view of (i), M f is a premean, it is reflexive.
In the case when the function (4) is strictly decreasing in one of the variables, arguing similarly, we show that F is strictly decreasing. Consequently, the function M f , being a composition of two strictly decreasing functions, is increasing. In both cases the continuity of M f is obvious. Now (ii) follows from Remark 1.

Definition 1.
Under the assumptions of Theorem 1 (ii), the function M f : I k → I is referred to as an additive-type quasi-Cauchy quotient mean generated by f (or of generator f ).
To see this assume, on the contrary, that for some monotonic function Hence, by induction, we get If f were bounded in a right vicinity of 0, then, letting n → ∞, we would get f (x) = 0 for all x > 0.
Thus lim x→0 f (x) = ∞. Consequently, f must be decreasing and 2f (x) = 1 2 < x for all x > 0 is invalid, so M is not reflexive. Since a positive continuous solution of this iterative equation depends on an arbitrary function (see [6]), the condition of monotonicity of f in Remark 5 is essential.
The equality M f = M g leads to the functional equation which is a special case of a more general functional equation in which ϕ, h and g are unknown.
Part 1: Additive quasi-Cauchy quotient for exponential functions Applying Theorem 1 for the exponential function of the form f (x) = p x we get is an additive type quasi-Cauchy quotient mean generated by the exponential and setting The function F maps R onto (0, ∞) , it is strictly increasing if p > 1 and strictly decreasing if 0 < p < 1; so F satisfies all the conditions of Theorem 1(ii). To show that the function is strictly monotonic with respect to the first variable, let us fix arbitrarily where a := r x2+···+x k and b := r x2 + · · · + r x k are some positive real numbers. The function ϕ, being the composition of the strictly increasing homographic function (0, ∞) u −→ au u+m and the exponential function R u −→ p u , is strictly increasing if p > 1, and strictly decreasing if 0 < p < 1. It follows that the function M p is strictly monotonic with respect to the first variable.
The symmetry of M p implies that it is strictly monotonic with respect to each variable, and, consequently, all the assumptions of Theorem 1 (ii) are satisfied. Since applying Theorem 1(ii), we get and, after an easy calculation, by the L'Hospital rule, we obtain e x1 ln p + · · · + e x k ln p (x 1 + · · · + x k ) − e x1 ln p x 1 + · · · + e x k ln p x k e x1 ln p + · · · + e x k ln p To calculate the limit at 0, assume first that x 1 , . . . , x k ∈ R are such that Making use of the above formula, we have By the L'Hospital rule, taking into account that the numbers x 2 − x 1 , . . . , x k − x 1 are positive, and letting p → 0, we get Now assume that x 1 , . . . , x k ∈ R are such that, for some l ∈ N, 1 < l < k, In this case we have and letting p → 0, we get so, generally, Now the symmetry of M p implies that for all x 1 , . . . , x k ∈ R, and, of course, is a symmetric mean. We omit the analogous reasoning in the calculations of lim p→∞ M p .
Remark 7. Proposition 1 implies that for all k ≥ 2 and p > 0, p = 1, so M p is exponentially conjugate to B k , the Beta-type mean defined in the introduction. In the case k = 2 we get M p (x, y) = log p H (p x , p y ), where H : (0, ∞) 2 → (0, ∞), given by H (x, y) = 2xy x+y is the harmonic mean. Thus M p is exponentially conjugate to the harmonic mean. Proof. Since, after simple calculations, for all x, y ∈ R, the mean M 1 is M 1 q , M q -invariant, and the result follows from [7]. (ii) the premean M ln is not a mean in any of the intervals (α, ∞) where α ≥ 1.
Considering this function we conclude that ϕ has exactly one global minimum at a point c ∈ (a, b) ⊂ (1, ∞) and, consequently, the function ϕ : (1, ∞) → (0, ∞) is strictly decreasing in the interval (1, c] and is strictly increasing in [c, ∞) , for some c ∈ (1, ∞) . This property implies that for arbitrarily fixed which proves that condition (2) of Theorem 1 is satisfied. Since by Theorem 1, the function is a k-variable premean in (1, ∞) .
Since, for each j ∈ {1, . . . , k}, Of course, it follows that M f := M ln is not a mean in any closed under addition subinterval J of (1, ∞).

Part 3: Additive quasi-Cauchy quotient for power functions
It turns out that Remark 8. There are no additive-type quasi-Cauchy quotient premeans generated by power functions.
Indeed, take an arbitrary k ∈ N, k ≥ 2, an interval I ⊂ (0, ∞) , closed under addition, and a power function f (x) = x p , where p ∈ R, p = 0. Since by (1), x ∈ I, the function F is constant, which shows that there does not exist an additivetype quasi-Cauchy quotient mean generated by f .

Exponential quasi-Cauchy quotient
Theorem 2. Let I ⊂ R be an interval that is closed under addition, i.e. I + I ⊂ I; f : I → (0, ∞) be a function; k ∈ N, k ≥ 2; and let F : I → (0, ∞) be given by

Then (i) if F is one-to-one and
then the function M f : is a correctly defined symmetric k-variable premean in I; (ii) if f is continuous and the function is strictly increasing (decreasing) in one of the variables, then F is strictly increasing (decreasing) and the function M f defined by (7) is a continuous symmetric strictly increasing k-variable mean in I.
Proof. Analogous as in Theorem 1.

Definition 2.
Under the assumptions of Theorem 2 (ii), the function M : I k → I is referred to as an exponential-type quasi-Cauchy quotient mean generated by f (or, of generator f ).
which for k = 2 reduces to which is a special case of the functional equation where ϕ, h and g are unknown.

Part 4: Exponential quasi-Cauchy quotient for additive functions
Applying Theorem 2 we obtain where B k is the Beta-type mean given by .

Remark 10.
The mean M f coincides with the Beta-type k-variable mean obtained in [4]. For k = 2, the mean M is the classical harmonic mean.
is the exponential-type quasi-Cauchy quotient mean M f generated by the logarithmic function f = log p , so it does not depend on p. Moreover, in case k = 2, M (x, y) = exp (ln x) (ln y) 2 ln (x + y) 4 ln (x + y) (ln x) (ln y) ln 2 + 1 + 1 , x,y > 1.
Proof. Take k ∈ N, k ≥ 2, I = (1, ∞), and the function f = log p . Consider first the case p > 1. By (5) we have F is strictly decreasing and maps (1, ∞) onto (0, ∞). Since the function (8): is decreasing in each variable, (that is easy to verify), by formula (7) of Theorem 2(ii), we obtain the mean Since we do not have the effective form of F −1 , we get which, in particular, shows that M is independent of the choice of p.
If p ∈ (0, 1) the proof is similar, so we omit it. The result in the case k = 2 follows from the fact that one can easily determine the effective form of F −1 .
where B k is the Beta-type mean.

Logarithmic quasi-Cauchy quotient
Theorem 3. Let I ⊂ R be an interval that is closed under multiplication, f : I → (0, ∞) be a function; k ∈ N, k ≥ 2; and let F : I → (0, ∞) be given by

Then (i) if F is one-to-one and
then the function M f : is a correctly defined symmetric k-variable premean in I; (ii) if f is continuous and the function is strictly increasing (decreasing) in one of the variables, then F is strictly increasing (decreasing) and the function M f defined by (11) is a continuous symmetric strictly increasing k-variable mean in I.
Proof. Analogous as in Theorem 1.

Definition 3.
Under the assumptions of Theorem 3 (ii), the function M f : I k → I is referred to as a logarithmic-type quasi-Cauchy quotient mean generated by f (or of generator f ). Remark 11. Put G (x) := g(kx) kg(x) and ϕ := G • F −1 . Now the equality M f = M g leads to the functional equation which is a special case of a more general equation with unknowns ϕ, h and g.
where B k is the Beta-type mean.
so, by symmetry, it is strictly increasing in each of the variables. Now Theorem 3(ii) implies that the function M f : (1, ∞) k → (1, ∞) given by is the desired mean M p := M f . Since, in general, we do not have the effective formula for F −1 , we get Hence, making use of the L'Hospital rule, for all x 1 , . . . , x k > 1, we get Now the properties of the function t k − t imply that M 0 ( x 1 , . . . , x k ) = lim p→0 M p exists, M 0 is a mean as the limit of means, and [M 0 (x 1 , . . . , x k )] k − M 0 (x 1 , . . . , x k ) = x 1 · · · · · x k − x 1 + · · · + x k k , x 1 , . . . , x k > 1.
Remark 20. The mean M is logarithmically conjugate to the mean B k . The above proposition remains true on replacing the interval (1, ∞) by (0, 1).

Part 12: Multiplicative quasi-Cauchy quotient for additive functions
Remark 21. The additive functions do not generate any multiplicative-type quasi-Cauchy quotient means.

Acknowledgements
The author is grateful to anonymous referees for their careful corrections to and valuable comments on the original version of this paper.
Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/.
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.