Invariance of means

We give a survey of results dealing with the problem of invariance of means which, for means of two variables, is expressed by the equality $$K\circ \left( M,N\right) =K$$K∘M,N=K. At the very beginning the Gauss composition of means and its strict connection with the invariance problem is presented. Most of the reported research was done during the last two decades, when means theory became one of the most engaging and influential topics of the theory of functional equations. The main attention has been focused on quasi-arithmetic and weighted quasi-arithmetic means, also on some of their surroundings. Among other means of great importance Bajraktarević means and Cauchy means are discussed.


Introduction
The idea of a mean is as old in human cognition as that one of a number. Given quantities x 1 , . . . , x p one can intuitively look for a mean of them as any number M (x 1 , . . . , x p ) lying somewhere between the extreme values of x 1 , . . . , x p : (1.1) means can be found also in Hardy, Littlewood and Pólya [74], in the book [20] by Borweins and the survey [52] by Daróczy and Páles; see also the quite recent book [153] by Gh. Toader and Costin. For any x, y ∈ (0, +∞) we have  for all x, y ∈ (0, +∞) (cf. [20,64,65] and [52]). The above equality shows a rather surprising connection of invariant means and mean iterations to elliptic integrals.
Using iterates of the map (A, G): (0, +∞) 2 → (0, +∞) 2 we can rewrite the convergence of Gaussian recurrences (2.1) in the form During the last 50 years this procedure was considerably generalized by a number of mathematicians. Some important steps to this aim were made by Lehmer [100] in 1971, Schoenberg [147] in 1982, and Foster and Phillips [63] in 1984. Their ideas were followed and extended by Borweins in the book [20] three years later. A percipient reader can find the following result there. respectively, which is impossible. Consequently, x 1 = . . . = x p and we are done.
Recently, the Generalized Gauss Algorithm has been extended to means depending on parameter (see [84]). Given an interval I, a positive integer p and a non-void set Ω, a function M : I p × Ω → I is called a parametrized mean on I if M (·, ω) is a mean for all ω ∈ Ω. The main result of [84], viz. Theorem 3.5, makes use of iterates (M 1 , . . . , M p ) n of the mapping (M 1 , . . . , M p ) : I p × Ω → I p . They are defined according to the following definition coming from the paper [14] by Baron and Kuczma (see also [60] by Diamond for a parallel notion introduced independently).
Theorem 3.5 of [84] generalizes Theorem 2.1 to the case when the set Ω of parameters is a compact topological space. That assumption is essential for the validity of Theorem 3.5 as follows from Example 3.4 presented in [84].

Quasi-arithmetic means
They as well as more general weighted quasi-arithmetic means constitute classes naturally extending the arithmetic mean A. It seems that the idea of a quasi-arithmetic mean was formed in [94] by Knopp already in 1928. Then the notion was formally introduced independently and almost simultaneously by Kolmogoroff [97], Nagumo [137] in 1930 and by de Finetti [62] a year later. Given an interval I we denote by CM(I) the class of continuous strictly monotonic functions mapping I into R. A mean M : In what follows the quasi-arithmetic mean generated by ϕ will be denoted by A ϕ . Observe that the mean A ϕ is conjugated to the arithmetic mean by AEM ϕ. Namely, if n ∈ R is fixed and we consider A ϕ as a mean in n variables: A ϕ : I n → I, then A ϕ = ϕ −1 • A • (ϕ, . . . , ϕ) where we take n copies of ϕ in the parentheses. There is at least one name more, namely that of Chisini, when thinking about the origins of the notion of quasi-arithmetic mean; de Finetti based his paper [62] mainly on some ideas of Chisini [29] presented in 1929, that is still a year before papers [97] and [137] by Kolmogoroff and Nagumo, respectively. Nowadays quasi-arithmetic means are sometimes also called Kolmogoroff means.
The classical means of the ancient world: the arithmetic mean A, the geometric mean G and the harmonic mean H are quasi-arithmetic. Indeed, since There is a vast literature dealing with quasi-arithmetic means. First of all, chronologically, we mention the paper [74] by Hardy, Littlewood and Pólya. Next, the late forties has a rich bibliography relating to quasi-arithmetic means: [1,5,76,[86][87][88]93,146,152] (see also [30] and [31]). Some newer results have been described for instance in [96,107,[142][143][144][145]. The last one deals with the speed of convergence of Gauss iterations for a class of mean-type mappings built with some quasi-arithmetic means. Also some books deal with this kind of means: [2] by Aczél, [22] by Bullen,Mitrinović and Vasić,[4] by Aczél and Dhombres, finally [21] due to P.S. Bullen.
The quasi-arithmetic means considered in the present paper are in two variables. The below famous characterization of quasi-arithmetic means was proved by Aczél (cf. [1] and [2, 6.4.1]): The following useful notion us allows to simplify formulations and proofs of results dealing with means having function generators, so, in particular, concerning the problem of invariance in the class of quasi-arithmetic means. Given a set X we say that functions ϕ : X → R and ψ : X → R are equivalent or ϕ is equivalent to ψ if there are numbers a ∈ R \ {0} and b ∈ R such that or, simply, ϕ ∼ ψ. Clearly, ∼ is an equivalence relation in the set R X of real-valued functions defined on X.

Theorem 3.2. Let I be an interval and ϕ, ψ ∈ CM(I). Then
We will refer to this result while studying the invariance problem in the class of quasi-arithmetic means.

The Matkowski-Sutô problem
Fix a non-trivial interval I ⊂ R. During the 5th International Conference on Functional Equations and Inequalities held in Muszyna-Z lockie (Poland) in 1995 Matkowski asked about all functions ϕ, ψ ∈ CM(I) such that the pair (ϕ, ψ) satisfies the functional equation This problem was published in [108] but only in 1998. A year later he gave the following partial answer (see [109,Theorem 1] with some a ∈ R \ {0}. Observe that Eq. (3.1) can be rewritten in the equivalent form for every x, y ∈ I, with some a ∈ R \ {0}.
The regularity assumption about the generators ϕ and ψ is not quite natural since the formulation of the original problem does not involve regularity conditions at all. But it was very useful in Matkowski's proof. The crucial tool in his reasoning is the following fact.
Simultaneously, Hungarian colleagues were also working on Matkowski's problem. First of all they discovered the elderly and forgotten two-part paper [149,150] written by Sutô and published in the Tôhoku Mathematical Journal in 1914. It deals with a number of functional equations, in particular with Eq. (3.1). Sutô's result (see [150]) requires stronger regularity assumptions than those made by Matkowski in Theorem 3.3; it reads as follows. It was published only in 2001 in their paper [49], where the so-called conjugate arithmetic means, that is means on I of the form with ϕ ∈ CM(I), have been studied (see also subsection 2.5.b). The main result of [49], that is Theorem 1, provides all continuously differentiable functions ϕ generating conjugate arithmetic means which are simultaneously quasiarithmetic. Making use of this the authors deduced Theorem 3.7. Its immediate proof was presented by Daróczy and Páles in the article [52] a year later. The following extension theorem, proved by Daróczy, Maksa and Páles in [47] turned out to be a useful tool there (see also [52,Theorem 3.14]), and in the subsequent research which finally solved the Matkowski-Sutô problem (see [52,Sec. 4] Making some simple calculations one can show that when proving Theorem 3.7 we can confine ourselves to the case when both functions ϕ, ψ are continuously differentiable. The next step is to observe that ϕ (x) = 0 and ψ (x) = 0 for x's running through a non-trivial subinterval of I. In fact the following much more general fact holds true (see [52,Theorem 4.8]). Proposition 3.9. Let ϕ, ψ ∈ CM(I) and assume that the pair (ϕ, ψ) satisfies Eq. (3.1). Then there exists a non-trivial interval K ⊂ I on which ϕ and ψ are differentiable and ϕ (x) = 0 and ψ (x) = 0 for all x ∈ K.
The above theorem was an important tool while solving the Matkowski-Sutô problem in the general form. It allows us to make the step from the continuity of a solution to its differentiability on a subinterval. We postpone the description of next steps in proving Theorem 3.7 for a moment to point out main facts resulting in a proof of Proposition 3.9.
First of all, making use of Lebesgue's theorem on differentiating monotonic functions almost everywhere (with respect to the Lebesgue measure) and the fact that the Lebesgue integral of the derivative of an absolutely continuous function over an interval is the increment of the given function, one can come to the following important fact (see [52,Theorems 4.1 and 4.3]).
Since the set of all t's from J satisfying condition (3.7) is a closed subset of J it can be deduced from Proposition 3.11 that both ϕ and ψ are differentiable on a non-trivial interval K ⊂ I. Moreover, it follows from Proposition 3.10 that ϕ (x) = 0 and ψ (x) = 0 whenever x ∈ K. This gives the assertion of Proposition 3.9. Now coming back to the sketch of the proof of Theorem 3.7, we may assume that the functions ϕ and ψ are continuously differentiable and, in addition, their derivatives do not vanish in the interval K. Since the derivative of any function defined on an interval has the Darboux property, we may assume that, in fact, ϕ (x) > 0 and ψ (x) > 0 for all x ∈ K. By virtue of Proposition 3.8 it is enough to prove that either condition (3.5), or (3.6) with some a ∈ R \ {0}, holds. This, however, can be easily obtained having the following series of lemmas (see [52,Theorem 3.7, and Lemmas 3.8 and 3.9], also [49,Lemma 3]).
Actually this results was formulated under the additional assumption of the continuity of ϕ and ψ which, however, was not used in its proof (cf. [52,Theorem 3.7]).
The last lemma deals with the equation containing only one unknown function.

Invariance of means 813
A little bit earlier the Hungarian group, working hard to remove the regularity assumption in Theorem 3.7, also tried to steer the research in a different direction. This other approach was connected with the notion of strict comparability of means (cf. [74] also [  First of all observe that, according to Propositions 3.9 and 3.8 , we may assume that the functions ϕ and ψ are differentiable and have nonvanishing derivatives. The derivative of any function defined on an interval has the Darboux property. Thus, replacing if necessary ϕ by −ϕ and/or ψ by −ψ (cf. Theorem 3.2), we may additionally assume that ϕ (x) > 0 and ψ (x) > 0 for all x ∈ I. Then, putting f = ϕ • ϕ −1 , g = ψ • ϕ −1 and applying Lemma 3.12, we see that the pair (f, g) is a solution of Eq.  While proving this result one may assume that both f and g are constant on no non-trivial subinterval of J. Since derivatives are of Baire class 1, so are elements of the class D(J). Thus, by virtue of Baire's theorem (cf. for instance, [138,Theorem 7.3], the set of all points of continuity of the function g is a dense G δ subset of the interval J. Using this fact one can show the existence of such u 0 , v 0 ∈ J that g is continuous at u 0 , v 0 and g (u 0 ) = g (v 0 ). Therefore, since the pair (f, g) satisfies Eq. (3.8), we see that for u and v running through neighbourhoods U and V of u 0 and v 0 , respectively. Now, using one of the results of Járai, viz. [77,Theorem 8.6], important in proving regularity, one can prove the continuity of f on a non-trivial interval. Now, making use of Proposition 3.17 we see that the function ϕ • ϕ −1 is continuous on a non-trivial subinterval of ϕ(I). Consequently, ϕ is continuously differentiable on a non-trivial subinterval of I, and thus the assertion of Theorem 3.16 follows from Theorem 3.7.
We complete this subsection with the below reformulation of Theorem 3.16 in the language of the invariance of means. It also generalizes Theorem 3.4.

Some supplementary remarks
At the very beginning we solve the problem of invariance in the class of quasiarithmetic means. The below result is an almost immediate consequence of Theorem 3.18.

10)
if and only if either for every x, y ∈ I, with some a ∈ R \ {0}.
Equation (3.10) can be seen as that expressing the generalized Matkowski-Sutô problem in the class of quasi-arithmetic means. Clearly, in different classes of means the answer to this problem can vary. Among quasi-arithmetic means especially important are power means or Hölder means which, in fact, constitute the class of all homogeneous quasi-arithmetic means (see [74]). Given a real number p we denote by H p the power mean on the interval (0, +∞), defined by Vol. 92 (2018)

Invariance of means 815
So H p = A hp , where h p : (0, +∞) → R is given by   [71] by G lazowska, the second present author and Matkowski, which should be also mentioned. There the authors determined all quasi-arithmetic means A ϕ , A ψ generated by twice continuously differentiable ϕ, ψ, and the real numbers r, s such that Actually we may assume that r + s = 1. Indeed, even more generally: if M and N are means on a common non-trivial interval I and rM + sN is a mean with some r, s ∈ R, then, by the reflexivity of the means, we have for all x ∈ I, whence r + s = 1 as I is non-trivial. So, in fact, we deal with the equation for all x, y running through the given interval. Observe that putting here r = 1/2 we come to Eq. (3.1) completely solved in the present section.
For some applications of the results presented in Section 3.3, both in the theory of means and in economy, as well as for some further facts and open problems concerning invariance in the class of quasi-arithmetic means, the reader is referred to [52,Chap. 5].

Solution in the class of weighted quasi-arithmetic means
Given an interval I, a function ϕ ∈ CM(I) and positive numbers p 1 , . . . , p n summing up to 1 we define the weighted quasi-arithmetic mean A ϕ (p1,...,pn) : I n → I putting for all x 1 , . . . , x n ∈ I. The function ϕ is called its generator and p 1 , . . . , p n are the weights of the mean. Clearly, the arithmetic mean A ϕ in n variables is weighted with p 1 = . . . = p n = 1/n. In what follows we focus on weighted quasi-arithmetic means in two variables. Then instead of A ϕ p,1−p we write A ϕ p , so given ϕ ∈ CM(I) and p ∈ (0, 1) we have for all x, y ∈ I. The below extension of Theorem 3.2 to weighted quasi-arithmetic means will be useful in the next discussion (see [2, Sec. 6.4, Theorem 2], also [106]). The problem of invariance in the class of weighted quasi-arithmetic means on the given interval I is to look for all functions ϕ, ψ, χ ∈ CM(I) and numbers p, q, r ∈ (0, 1) such that the mean A χ r is the Gauss composition of A ϕ p and A ψ q : Considering the definition of Gauss composition and the form of weighted quasi-arithmetic means one can write Eq. (3.14) in the following equivalent form: Observe that putting here χ = id| I and p = q = r = 1/2 we come to Eq. (3.1). It seems that it was paper [53] by Daróczy and Páles where a first step in solving the general form of (3.15) was made. There the authors determined all continuously differentiable, with nonvanishing derivatives functions ϕ, ψ ∈ CM(I), satisfying the equation in the case when r ∈ (0, 1) \{1/2} (see [53,Theorem]). Since (3.16) for r = 1/2 is simply (3.1), it follows from Theorem 3.16 that in the case r ∈ (0, 1) we are done in the considered class of functions. Clearly, Eq. (3.16) is a particular case of (3.15) with χ = id I and p = q = r.
In the same year, 2003, Daróczy and Páles gave the complete solution of Eq. (3.16) in the class CM(I), proving the following result in [54]. The set of solutions in this class is exactly the same as that described in [53]. Consequently, it follows from Theorems 3.16 and 3.22 that Eq. (3.16), although formally more general than (3.1), admits the same solutions (ϕ, ψ). One of the tools allowing us to pass from the result of [53] to Theorem 3.22 was an extension theorem for solutions of Eq. (3.16) obtained by Daróczy, Hajdu and Ng [43] in 2003.
Another particular case of Eq. (3.15) different from (3.16), viz. the equation was investigated by Burai in [23]. Clearly, we come to (3.17) putting χ = id| I , q = 1 − p and r = 1/2 in Eq. (3.15). On the other hand (3.17) with p = 1/2 becomes Eq. (3.1). It turns out that also this problem has the same set of solutions as the cases considered previously. Its detailed description is given in the following result (see [23,Theorem 6] A key role in the proof of Theorem 3.24 is played by the below description of conditionally homogeneous weighted quasi-arithmetic means (see [44], also [85, Proposition 1]).

The mean A σ s is conditionally homogeneous:
A σ s (tx, ty) = tA σ s (x, y) for all x, y ∈ I and t ∈ (0, +∞) with tx, ty ∈ I if and only if either This result generalizes the classical theorem stating that power means H p , where p ∈ R, are the only homogeneous quasi-arithmetic means on (0, +∞) (see [74]). For further generalizations the reader is referred to paper [25] by Burai and the first author, where the conditional homogeneity of Makó-Páles means of the form where ϕ ∈ CM(I) and μ is a probability Borel measure on [0, 1], is examined (see [25,Theorem 3.1]). This class is a common extension of the classes of weighted quasi-arithmetic means and Lagrangian means. For more information on Makó-Páles means the reader is referred to Subsection 6.1, which is devoted mainly to the invariance of the arithmetic mean A as well as the geometric mean G with respect to a pair of Makó-Páles means.
Besides Proposition 3.25 the below three analogues of Lemmas 3.12-3.14 play a crucial role while proving Theorem 3.24 (see [85, In the case when p = q = 1/2 Eq. (3.19) becomes (3.8), so Lemma 3.26 generalizes Lemma 3.12. The authors of [85] assumed in Lemma 2 there that ϕ and ψ are continuous but this is superfluous (cf. also the comment just after Lemma 3.12). The next result, although deals with solutions of Eq. (3.19), which is more general than Eq. (3.8), does not extend Lemma 3.13. This is because of the stronger assumption imposed on solutions here.
Lemma 3.27. Let p, q ∈ (0, 1). If f and g are continuously differentiable functions mapping an interval J into (0, +∞) and the pair (f, g) satisfies equation (3.19), then there exists a c ∈ (0, +∞) such that No doubt the last lemma, originating from paper [85], has the most involved argument. It deals with the equation 21) which, after setting p = q = 1/2, reduces to Eq. (3.9). Therefore the below lemma generalizes Lemma 3.14.
Finally the problem of invariance in the class of weighted quasiarithmetic means, i.e. Eq. (3.14), was completely solved in [78]    To prove this result the argument used by Daróczy and Páles in validating Theorem 3.16 has been extended substantially. Below we formulate Proposition 3.30, Lemma 3.31 and Proposition 3.32, the main tools of [78]. They are generalizations of Proposition 3.17, Lemma 3.13 and Proposition 3.8, respectively (cf. [78,). The form of Eq. (3.18) is significantly more complicated than that of Eq. (3.1), and thus the proofs are now essentially longer and more involved. Also methods elaborated by Járai in monograph [77] are used to a larger extent than in the proof of Theorem 3.16.  there. The proof of Lemma 3.31 given in [78] runs in another way than that of Lemma 3.13 presented in [52]. If p = q then Eq. (3.19) takes the form (which is (3.8) when we put p = 1/2) and the assertion follows from Theorem 2 by Daróczy and Páles as published in their paper [55]  In the special case when p = q = r in Eq. (3.18) the assertion of Proposition 3.32 follows immediately from [42,Theorem 4] where the extension theorem was proved for the equation The proof of Theorem 3.29 presented in [78] starts with showing that the functions ϕ, ϕ −1 , ψ, ψ −1 are locally Lipschitzian and their derivatives do not vanish wherever they exist. Since they have the Darboux property we may assume without loss of generality that they take positive values only. The next step is to prove that ϕ, ψ are differentiable on a non-trivial interval I 0 ⊂ I. This initial part of the proof takes the pattern of the argument used by Daróczy and Páles while proving Theorem 3.16. It requires only a bit of care as the complexity of Eq. (3.18) is higher than that of Eq. (3.1). Now, defining f : J 0 → (0, +∞) and g :  (3.19) in the form of (3.21). Then, making use of Lemma 3.28 and the definition of f , we obtain the desired form of the function ϕ, and then also ψ, on a non-trivial subinterval of I. To complete the proof it is enough to apply Proposition 3.32.
We complete this Subsection with the following corollary from Theorem 3.29 providing the form of all weighted quasi-arithmetic means satisfying Eq. (3.14). This gives the full answer to the problem of invariance in the class of these means and generalizes Theorem 3.19.

Generalized weighted quasi-arithmetic means in the sense of Matkowski
This generalization seems to be especially important and promising for further research. Given an interval I and strictly increasing (or strictly decreasing) functions ϕ, ψ ∈ CM(I) we define the mean A [ϕ,ψ] : I 2 → I by the formula It was introduced by Matkowski [114] in 2003. The first invariance results were obtained six years later by Matkowski and Volkmann in the paper [132] and by Baják and Páles in [6] (cf. also [124]). Taking any ϕ ∈ CM(I) and p ∈ (0, 1) we see that , that is the considered notion extends that of weighted quasi-arithmetic mean. For this reason A [ϕ,ψ] is called generalized weighted quasi-arithmetic mean with the generators ϕ and ψ. In fact, while defining the mean A [ϕ,ψ] , it is enough to assume less, namely that the functions ϕ, ψ : I → R are such that ϕ(I)+ψ(I) ⊂ (ϕ + ψ) (I) and ϕ + ψ ∈ CM(I).
Many different properties of generalized weighted quasi-arithmetic means have been extensively studied by Matkowski in his papers [119,128,130] and [131]. Most of them deal with a natural generalization of A [ϕ,ψ] to the mean A [ϕ1,...,ϕn] in n variables, where n ≥ 2 is a fixed integer. Given functions ϕ 1 , . . . , ϕ n ∈ CM(I) of the same type of monotonicity we put To formulate a result solving the equality problem for generalized weighted quasi-arithmetic means we extend the notion of equivalence of functions described in Subsection 3.1. Given a set X we say that n-tuples (ϕ 1 , . . . , ϕ n ) and (ψ 1 , . . . , ψ n ) of real-valued functions defined on X are equivalent or (ϕ 1 , .
Observe that ∼ is an equivalence relation in the set of functions mapping X into R n .  [13,35,41,50].
What concerns invariance problems the list of results is so far rather short. The first attempt was made by Matkowski and Volkmann [132] in 2008. It answers the question on the invariance of the arithmetic mean A with respect to the pair A [ϕ,ψ] , A [ψ,ϕ] considered on a fixed interval I. Theorem 3.35. Let ϕ, ψ ∈ CM(I) be functions of the same type of monotonicity. Then the pair (ϕ, ψ) satisfies the equation if and only if expressing the invariance of the mean A with respect to the mean-type map- ψ2] ; to see this it is enough to put ϕ 1 = ψ 2 = ϕ and ϕ 2 = ψ 1 = ψ. The study of Eq. (3.24), unlike its special case (3.23), has turned out to be more complicated. The following result was proved by Baják and Páles in the paper [6].
for all x, y ∈ I.
The problem of relaxing the regularity assumption in Theorem 3.36 is still open.
The next result, proved by Matkowski in [130], provides an interesting invariance formula in the class of generalized weighted quasi-arithmetic means (see [130,Theorem 2] In 2015, Matkowski and Páles gave a characterization of generalized quasiarithmetic means which involves this fact and a generalized bisymmetry equation (see [131,Theorem 5]).

Around weighted quasi-arithmetic means
a. We begin with a recent paper [92] by Kahlig and Matkowski where the weighted arithmetic, geometric and harmonic means (but not only this one) are the main heros. Given a number p ∈ (0, 1) we define the means A p : R 2 → R, G p : (0, +∞) 2 → (0, +∞) and H p : (0, +∞) 2 → (0, +∞) by Moreover, they characterized those (p, q)'s for which N p,q is a mean. In other words, we know all pairs (p, q) for which the mean M p has a K p -complementary mean, that is N p,q , and the form N p,q .
In the rest of Subsection 3.6 we deal with some generalizations of weighted quasi-arithmetic means. Some others will be discussed later. In particular, in Subsection 6.1 the so-called Makó-Páles means, which are a common generalization of both weighted quasi-arithmetic means as well as Lagrangian means, are considered. b. It seems that the notion of a conjugate mean was originally introduced in paper [35] by Daróczy in 1999 who was inspired by Matkowski [108] (see also [50] by Daróczy and Páles). Given a mean L on an interval I a mean M : I 2 → I was called by him L-conjugate if there exists a function ϕ ∈ CM(I) such that x,y∈ I.
Later, in 2001, Daróczy and Páles generalized it setting parameters into the above equality (see [49], also [56] and [39]). Namely, M is said by them to be L-conjugate, if , y))) , x,y∈ I, with some function ϕ ∈ CM(I) and numbers p, q ∈ [0, 1]. Putting here p = q = 1 we come to the previous version of the definition. The mean M of the above form is denoted by L (p,q) ϕ . The function ϕ is called its generator and the numbers p, q are its weights. Observe that L (0,0) ϕ = L and L (p,1−p) ϕ = A ϕ p for an arbitrary mean L and all ϕ ∈ CM(I), p ∈ (0, 1). Notice also that the equation L (p,q) ϕ = A ψ ν with L = A ϕ μ was completely solved in [83] with no regularity assumption imposed on the generators ϕ and ψ.
The invariance of the arithmetic mean A with respect to the pair A (1,1) ϕ , A (1,1) ψ of two A-conjugated means, that is the equation (3.25) was considered by Daróczy [36] in 2000. He found all pairs (ϕ, ψ) of twice continuously differentiable functions satisfying Eq. (3.25). An extension theorem for solutions of Eq. (3.25) was proved by Hajdu [73] in 2002. She showed that each solution (ϕ, ψ) of (3.25) can be uniquely extended from any non-trivial interval K ⊂ I to the whole I. Fifteen years later, in 2015, Sonubon and Orankitjaroen [148] proved the following result trying to make a progress in this research. Theorem 3.39. Let ϕ, ψ ∈ CM(I) be three times continuously differentiable and p, q, r, s, t ∈ (0, 1). Assume that p = q, p + q = 1, r = s, r + s = 1, ts = (1 − t)r, and either p + q = r + s, or p + q = 2(r + s). If the pair (ϕ, ψ) satisfies the equation with p = s = 1/3, q = r = 1/2 and t = 3/5. Therefore the problem of invariance of the arithmetic mean A with respect to pairs of parametrized A t -conjugated means is still open, even in the class of pairs of three times continuously differentiable functions. By the way, it is a curio that Theorem 2.1 from [148], which is the main tool of the proof of Theorem 3.39, is apparently recalled as Theorem 8 from [105] but it is cited improperly, in an incomplete form.
It is worth noting that some invariance problems for means are particular cases of the equality problem for conjugate means which is widely studied in the literature (cf., for example, [13,35,39,41,50,126]). If, for instance, ϕ, ψ ∈ CM(I) and (p, q) ∈ (0, 1) 2 satisfy the equation expressing the invariance of the arithmetic mean A with respect to the pair A ϕ p , A ψ q of weighted quasi-arithmetic means, then, putting In 2002 Daróczy and Páles [51] introduced the following other generalization of the notion of quasi-arithmetic mean. Given an interval I and a real number α ≥ −1, a mean M : I 2 → I is called quasi-arithmetic of order α if there exists a function ϕ ∈ CM(I) such that In such a case the mean M is denoted by A (α) ϕ , the function ϕ is called its generator and α its order. Observe that x,y∈ I, so means of order 0 coincide with quasi-arithmetic means and means of order −1 are conjugate quasi-arithmetic means. In [51] the authors studied the invariance of the arithmetic mean with respect to the pair of quasi-arithmetic means of fixed order α, that is the equation  As an almost immediate consequence we obtain the following description of all pairs A   [57] and [58]). Let I be a non-trivial interval. Given a function ϕ ∈ CM(I) and a parameter p ∈ (0, 1) we define the mean A * ϕ p : I 2 → I by

Theorem 3.41. Let at least one of functions ϕ, ψ ∈ CM(I) be continuously differentiable and α ∈ [−1, +∞). Then the arithmetic mean A is invariant with respect to the pair
in other words for all x, y ∈ I. Some fundamental properties of symmetrized quasi-arithmetic means were proved in another paper [58] by Daróczy and Páles. There the authors gave necessary and sufficient conditions for the comparison, equality and homogeneity of means of the form where M : I n → I is an arbitrary mean, ϕ ∈ CM(I) and ω 1 , . . . , ω n : I 2 → (0, 1); here A ϕ ωi is the weighted quasi-arithmetic mean generated by ϕ with function weight ω i : y))ϕ(y)) , x,y∈ I.
Putting here n = 2, M = A, and taking the constant weights ω 1 = p and ω 2 = 1 − p we come to the symmetrized weighted quasi-arithmetic mean A * ϕ p .

Invariance of means 829
The equality problem for means of form (3.29) has the following solution (see [58,Theorem 3.3]).
The detailed discussion of invariance for weighted quasi-arithmetic means is postponed to Section 3.
It seems that invariance problems for means of form (3.29) are, in general, complex. However, in the particular case of symmetrized weighted quasiarithmetic means the situation is much easier. In 2013 Burai proved what follows (see [24,Theorem 2]).

Theorem 3.43. Let ϕ, ψ ∈ CM(I) be two times continuously differentiable. Assume that at least one of ϕ, ψ is four times continuously differentiable and
Then the pair (ϕ, ψ) satisfies the equation if and only if either condition (3.2), or (3.3) with some a ∈ R \ {0}, holds.
The assumption (3.30) looks rather strange and it is not known if it is essential for the validity of Theorem 3.43. The proof is technical, tedious and comes down to solving some complicated differential equations derived on a nontrivial subinterval of I. Condition (3.30) allows us to exclude a hopeless case while studying one of these equations. To omit the exclusion (3.30) Burai needed to presume higher order regularity of the generators ϕ, ψ (see [24,Theorem 3] Another tool while proving both results is an extension theorem for solutions of Eq. (3.31) (see [24,Theorem 4]). On the one hand it allows us to spread the information obtained on the generators when solving the derived differential equations to the whole interval I. On the other hand, using this extension theorem, one can formulate Theorems 3.43 and 3.44 in a more general form where the regularity assumptions are imposed on the generators only on a subinterval of I.

Close to invariance
Some natural generalizations of the invariance of the Matkowski-Sutô problem lead to equations not expressing the invariance of the mean. Such is, for instance, the problem described by Eq. (3.13) studied in [71] by G lazowska, the second present author and Matkowski. The below result (see [71,Theorem]), published in 2002, provides all solutions (ϕ, ψ) of (3.13), consisting of functions which are regular enough. In what follows I denotes a fixed non-trivial real interval.
with some x 0 ∈ R \ I.
A year later Daróczy and Páles [55] extended the research to the equation It is interesting that in spite of adding a parameter to Eq. (3.13) and weakening the regularity assumptions the set of solutions does not enlarge. When proving Theorem 3. 46 Daróczy and Páles patterned after their argument used in [52] to solve the Matkowski-Sutô problem under the assumption of continuous differentiability (see Theorem 3.7 and the sketch of its proof presented in Subsection 3.2). Roughly speaking the procedure runs according to the following schedule: 1. The pair (f, g), where f = ϕ • ϕ −1 and g = ψ • ϕ −1 , satisfies the equation and f is a solution of the equation   Clearly, putting q = s = 1/2 in (3.35) we come to Eq. (3.31). The papers [40] and [59] provide the form of all solutions (ϕ, ψ) of Eq. (3.35) under the assumptions that the functions ϕ, ψ ∈ CM(I) are differentiable with nonvanishing derivatives and the parameters q, r, s are assumed to satisfy the following conditions: The invariance results of both papers have been generalized by Daróczy in [38] by relaxing both the regularity assumption and the conditions imposed on the parameters. Namely, he determined all functions ϕ, ψ ∈ CM(I) such that (ϕ, ψ) satisfies Eq. (3.35) assuming only that q ∈ (0, 1) \ 1 2 and r, s ∈ R \ {0, 1}.
It is interesting that the method, applied by him to solve the problem there, essentially differs from those used before. Roughly speaking it relies on Theorems 1 and 2 from [38] providing complete answers to the following two questions. Given a nonsymmetric weighted quasi-arithmetic mean M on I and nonzero real numbers α, β, γ, α = β, find necessary and sufficient conditions for the function N : to be (a) symmetric: symmetric, and bisymmetric: N(y, v)), x, y, u, v ∈ I (cf. also Aczél's characterization of quasi-arithmetic means contained in Theorem 3.1). When answering these questions Daróczy made use of results of paper [45] by him and Maksa.
In 2010 the first author of the present paper completed her research concerning the equation with some x 0 ∈ R \ I.
Observe that if I = R, i.e. there is no x 0 ∈ R \ I, then only cases (a), (b), (c) are possible. Briefly speaking Theorem 3.47 can be easily obtained using two crucial results. The first of them (see [81,Theorem 2]) provides all the possible forms of local solutions of (3.36). One of the main tools used in its proof is the following regularity theorem published in paper [80] by the first present author. Theorem 3.48. Let ϕ, ψ ∈ CM(I) and r, s ∈ R, p, q ∈ (0, +∞). If the pair (ϕ, ψ) satisfies Eq. (3.36), then there exists a non-trivial interval K ⊂ I such that the functions ϕ| K , ψ| K are infinitely many times differentiable with nonvanishing first derivatives.
Its proof follows ideas used by Daróczy and Páles while giving the ultimate answer to the Matkowski-Sutô problem in [52] (see Theorem 2.16 herein and the sketch of its proof) and their amplification applied in [78] to solving the problem of invariance in the class of weighted quasi-arithmetic means (see theorem 3.29 and part of Subsection 3.4 following it). As previously a key role is played by regularity improving results due to Járai (see [77,Theorems 8.6 and 11.6]).
Having determined all the local solutions of Eq. (3.36) the next problem is how to propagate the information about their forms to the whole interval I. The below extension result solves that problem. It generalizes Proposition 3.32, so consequently also 3.8. The following schedule of the procedure of determining local solutions of (3.36) is a further expansion of that applied by Daróczy and Páles in the proof of Theorem 3.46 (see [55]) and, originally, in solving the Matkowski-Sutô problem in [52], or by the first present author while proving Theorem 3.29 (see [78]). 1. The pair (f, g), where f = ϕ • ϕ −1 and g = ψ • ϕ −1 , satisfies the equation  with a ∈ R \ {0}, with ξ < I and ξ > I, respectively (weighted power means centered at ξ), with ξ < I and ξ > I, respectively (weighted geometric means centered at ξ). The result below is an immediate consequence of Theorem 3.47.
Remark 3.51. Recently, rather unexpectedly, Theorem 3.50 (or 3.47) has enabled us to solve the following problem which came from iteration theory.

Given a non-trivial interval I and a mean-type mapping
A complete answer to this problem was given in [70] by G lazowska and the authors of this survey.
The equality problem for Bajraktarević means with p 1 = . . . = p n , postulated for all n ≥ 2, was solved by Aczél and Daróczy in [3] already in 1963 (there Bajraktarević means are called generalized quasi-linear means). The comparison, and thus also the equality problem, in the general case was solved in [106] by Maksa and Páles. For some other rather partial results the reader is referred to [10,44,101,103] and [48]. The following result can be deduced from [106,Theorem 3]. if and only if ϕ ∼ ψ and p = q.
It seems that it is still an open problem if the regularity assumption made here can be removed. A characterization of Bajraktarević means of the form B ϕ,ω was given in [139]. The reader interested in one of the possible generalizations of such means should consult the paper [120], where B ϕ,ω 's are embedded in some one-and two-parameter families of means.
To discuss the invariance problem involving Bajraktarević means we start with a few results on the invariance of the arithmetic mean and some similar ones with respect to a pair or even a tuple of Bajraktarević means and some of their generalizations. Fix a real interval I. They were Domsta and Matkowski who studied it probably for the first time. In [61], assuming that I ⊂ (0, +∞), they solved the equation In fact, the assertion of the above result remains true if we assume that at least one of functions ϕ, ψ ∈ CM(I) is four times countinuously differentiable. Notice also that if ϕ(x) ∼ 1/x for all x ∈ I, then B ϕ,Id (x, y) = x + y x 1 x + y 1 x,y∈ I, that is B ϕ,Id is simply the arithmetic mean A. A more general invariance problem, namely the invariance of the arithmetic mean with respect to the pair B ϕ,ω , B ψ,ω of Bajraktarević means in two variables, was studied in [82]. A suitable equation has the form The next result (see [82,Theorems 3 and 4]) generalizes Theorem 4.2. Here the function ω is assumed to be a fundamental solution of Eq. (4.4), that is one of the functions given by A standard computation shows that the functions ϕ and ψ described in Theorem 4.4 generate again the arithmetic mean. It would be interesting to find a non-trivial pair B ϕ,ω , B ψ,ω , i.e. different from (A, A), such that the arithmetic mean is invariant with respect to B ϕ,ω , B ψ,ω . Quite recently Páles and Zakaria have studied the equation where f, g, h, k : I → R are continuous functions such that g, k do not vanish and the functions f/g and h/k are strictly monotonic, and s, t are positive numbers (see [141]). This is the equation expressing the invariance of the arithmetic mean with respect to the pair B (s,t) . Substituting ϕ := f/g, χ := g, ψ := h/k and ω := k we can rewrite (4.5) as Vol. 92 (2018)

Invariance of means 839
Taking here χ = ω and s = t we obtain Eq. (4.3). Notice, however, that the assumption s = t is imposed on the parameters s, t in [141], everywhere the solutions of (4.6) are determined, so Theorems 4.3 and 4.4 above cannot be deduced from Theorem 1 proved in the paper [141]. There, under the assumption that ϕ, ψ are four times continuously differentiable and s = t, the authors found all solutions (ϕ, ψ) of (4.6): roughly speaking ϕ and ψ are of the form where Eq. (4.6) is a particular case of the equation where λ, μ, ν : I 2 → (0, 1) are given weighted functions and the unknown functions ϕ and ψ are assumed to belong to the class CM(I); it is enough to take λ(x, y) = 1/2 and for all x, y ∈ I. Extending the notion introduced in Section 3.4 by putting λ(x, y))ϕ(y)) we can rewrite (4.7) in the form Observe that for λ : I 2 → (0, 1) defined by we have A ϕ λ = B ϕ,ω . In the special case when λ is constant or, more generally, A ϕ μ , A ψ ν -invariant: with χ = Id. The above equation extends (3.14) to the case of quasi-arithmetic means with function weights.
In what follows we say that a function λ : I 2 → R is k-times differentiable in the first variable on the diagonal if for any x ∈ I the function λ(·, x) is ktimes differentiable at x. The following result (see [79,Theorem 2]) allows us to reduce the problem of determining solutions of Eq. (4.7) to that of solving the differential Ricatti equation It is well known that, in general, it is hard to solve the Ricatti equation effectively. However, as it follows from (4.9), in some particular cases we are able to manage the situation. This is, for instance, the case when Then the Ricatti Eq. (4.9) becomes an algebraic quadratic equation or a linear differential one. In general, however, we are far from determining all the solutions of (4.7). If we take any pair (ϕ, ψ) satisfying (4.7), then Theorem 4.5 may give only the form of ϕ /ϕ . Integrating it twice we come to ϕ (and then also to ψ). It is usually difficult to verify if the obtained pair (ϕ, ψ) really satisfies Eq. (4.7). Some particular situations, when we are able to decide it, were described in [79]. Other problems deal with the invariance of Bajraktarević means with respect to pairs of some other means. One of them was solved in [125] by Matkowski. With no regularity assumptions the following result brings the form of all Bajraktarević means B [f,g] in two variables which are invariant with respect to the pair A f , A g of quasi-arithmetic means: (4.10) (see [125,Theorem 1]).
for all x, y ∈ I, that is for all x, y ∈ I.
If, in addition, f, g are regular enough, then one can prove a similar theorem dealing with the more general equation which expresses the invariance of B [f,g] with respect to the pair A f p , A g r of weighted quasi-arithmetic means (see [125,Theorem 2]).
We complete this section with a short report on the paper [123] by Matkowski. It seems that much more important than the invariance result (see [123,Theorem 3]) is the following notion of a generalized Bajraktarević mean proposed there. Given a positive integer k denote by σ k the shift of the set {1, . . . , k} mod k that is the permutation defined by and by σ i k , for i = 0, . . . , k − 1, the ith iterate of σ k : Assuming that k ≥ 2 and given functions ϕ ∈ CM(I) and ω 1 , . . . , ω k : I → (0, +∞) we put for all x 1 , x 2 ∈ I (cf. [123,Remark 3]

Invariance of means 843
Moreover, instead of B ω (p,1−p) we write B ω p . In the case when ω is a power function the mean B ω was considered by Gini already in 1938 (cf. [66]) and then, in a more general setting, by Beckenbach (see [15]).
Classical means, viz. the arithmetic, geometric and harmonic ones, serve as typical examples of Beckenbach-Gini means. Indeed, if ω is constant: ω(x) = c ∈ (0, +∞), then In turn, taking ω(x) = 1/ √ x for each x ∈ (0, +∞) we get for all x, y ∈ (0, +∞). Finally, putting ω(x) = 1/x for each x ∈ (0, +∞), we obtain for all x, y ∈ (0, +∞). The problem of invariance in the class of Beckenbach-Gini means leads to the question on triples (ω, α, β) of positive functions defined on the interval I and satisfying the functional equation (4.11) It seems that in general it is hard to answer this question. We start with the particular cases when B ω is one of the means A, G, H. Then we are able to determine all the pairs satisfying (4.11) with no regularity assumptions. The result below has been proved by Matkowski (see [113,). For some further results concerning invariance in families of weighted Gini means the reader is referred to the papers [33] and [34] by Costin and Gh. Toader where the method of series expansion is applied again.

Generalities
Fix an interval I ⊂ R and differentiable functions f, g : I → R such that g does not vanish and the function f /g is one-to-one. Then the Cauchy mean value theorem implies that the formula defines a function D f,g : I 2 → I which is a strict mean on I. We call it the Cauchy mean generated by f and g. Observe that if also f does not vanish, then D f,g = D g,f . The equality problem not easy to solve, was treated by Losonczi in [102] under rather restrictive assumptions concerning, among others, the regularity of the generators. These restrictions have been removed by Matkowski in [115]. There one can also find a simple argument showing that under the above assumptions imposed on f and g we have, in fact, f /g ∈ CM(I) (cf. [115,Remark 1]). There are some important subclasses of the class of Cauchy means. The first one consists of those of the form D f,id ; such a mean is called the Lagrangian mean generated by f and denoted by L f : for all x, y ∈ I. A classical example is the logarithmic mean L log : for all x, y ∈ (0, +∞). Another class of Cauchy means has been proposed by Stolarsky [151]. Remembering the definition (3.11) of the function h p : (0, +∞) → R, p ∈ R, and given real parameters p, q such that p 2 + q 2 > 0 define the Stolarsky mean E p,q on (0, +∞) by the equalities Additionally we put E 0,0 = G. In other words, we have if p = q and pq = 0, if p = q and q = 0, if p = q and p = 0, for all x, y ∈ (0, +∞), x = y. Observe that E p,q = E q,p for all p, q ∈ R. The means E p,0 , p ∈ R, are also called extended logarithmic means, whereas the mean E 1,1 is called identric. Notice that E 1,0 = L log is simply the logarithmic mean. Moreover, E p,−p is the geometric mean G and E 2p,p is the power mean H p , p ∈ R; in particular, E 2,1 = A, E 0,0 = G, and E −2,−1 = H. Studying invariance in the class of Cauchy means leads to difficult problems and, as it seems, there are no results on solutions (ϕ, ψ, f, g, h, k) of the equation in the general case; only some of its particular cases have been considered up to now. The equation where one of the generators of each Cauchy mean is a power function, serves as an example of such a situation. It has been studied by G lazowska in [69]. The main result proved there reads as follows (see [69,Theorem 2]).
for all x ∈ I, or there exists r ∈ {0, p} such that As it follows from for all x ∈ I 0 .
The second one (see [69, Lemmas 2 and 3]) has a long and complicated formulation. For that reason we present it in the particular case when q = −p.
for all x ∈ I 0 .
Lemma 5.5, especially in its full form (cf. [69, Lemma 2]), has a tedious highly computational proof making use of both Theorem 5.3 and Lemma 5.4, and is the main tool while proving Theorem 5.1.
Difficulties, while solving the invariance problem in the class of Cauchy means, can be essentially reduced when we confine ourselves to some of its special subclasses, viz. to Lagrangian means or Stolarsky means.

Lagrangian means
Before a discussion of the invariance questions for Lagrangian means we deal with the equality problem concerning them. It was answered by Berrone with some a ∈ R \ {0}.
As an immediate consequence of Theorem 5.7 we obtain its reformulation in terms of means. For each a ∈ R \ {0} we denote by L a the Lagrangian mean generated by the function I x −→ e ax : for all x, y ∈ I. Additionally we put L 0 = A. In the proof of Theorem 5.7 the following result improving the regularity of the generators f, g is useful (see [116,Theorem 1]). Making use of this regularity theorem one can reduce the problem of determining the solutions of (5.3) to solving the differential equation in a subinterval of I. The equation which expresses the invariance of the geometric mean G with respect to a pair L f , L g of Lagrangian means, is completely solved. We start with the research made by G lazowska in [67] under the assumption of the conditional homogeneity of the means L f and L g . Assuming that I ⊂ (0, +∞) we say that a mean M : I 2 → I is conditionally homogeneous if for all x, y ∈ I and t ∈ (0, +∞) such that tx, ty ∈ (0, +∞). All conditionally homogeneous Lagrangian means are listed below (see [67,Theorem 2]). We use the following denotation: x−y (iii) there exists p ∈ R such that L f = L [p] .
In the proof [110, Theorem 1] as well as some ideas of the proof of [85, Proposition 2] have been used. Making use of Theorems 5.10 and 5.6 , and some calculus of derivatives, G lazowska proved the theorem below which is the main result of [67].
Theorem 5.11. Assume that I ⊂ (0, +∞). Let f, g : I → R be differentiable functions with one-to-one derivatives f and g . Assume that at least one of the means L f , L g is conditionally homogeneous. Then the following statements are pairwise equivalent: (i) the pair (f, g) satisfies Eq. (5.5); (ii) f (x) ∼ 1 x 2 , x ∈ I, and g (x) ∼ 1 The next step in solving Eq. (5.5) was made in the paper [72] by G lazowska and Matkowski. There they resigned the assumption of conditional homogeneity of Lagrangian means and proved the following necessary condition for the L f , L g -invariance of the geometric mean (cf. [72,Theorem 4]). In [72] the functions f and g are actually assumed continuously differentiable. This is, in fact, superfluous since any function differentiable on an interval, with one-to-one derivative, is continuously differentiable there (cf. [116,Remark 1]).
In the case when log x 3 f (x) ∼ x − 4 9 , x ∈ I, (and of course also (5.6) holds) the mean L f cannot be expressed by elementary functions. For this reason it was hard to decide if the geometric mean G is, in fact, invariant with respect to the pair L f , L g . This problem remained unsolved until 2011 when G lazowska [68] showed that this is not the case. This was done by calculating partial derivatives of order 7 of L f and L g satisfying (5.5). Some of the calculations were made using Mathematica 4.1.
Summarizing, the final answer to the question on solutions of Eq. (5.5) can be formulated as follows (cf. Theorem 5.12 and [68, Theorem 3.1]). We conclude this section with a result of Matkowski [122] solving the equation D f,g • L f , L g = D f,g , (5.7) expressing the invariance of the Cauchy mean D f,g with respect to a pair of Lagrangian means generated by the same functions f and g. Here the means L a , a ∈ R \{0}, defined by (5.4) play an important role (see [122,Theorem 1]). In its original formulation there is a lack of the assumption that the functions f and g are one-to-one, needful to define the means L f and L g . It does not follow from the assumption that f /g is one-to-one: f = log and g = id serve as an example. On the other hand there is no need to assume that f , g are continuously differentiable. This is forced by the invertibility of f and g . In the proof the lemma below is very useful (see [122,Lemma 2]).

Stolarsky means
The invariance problem in the class of Stolarsky means relies on solving the equation Its particular case when p + q = 0, that is the equation (a, b, c, d) satisfying it. Some of them were formally losed. Nevertheless, as E p,q = E q,p for all p, q ∈ R, all the means E a,b and E c,d were finally determined. Below we present a reformulation of Theorem 2 from [19] made also in terms of means, which seems to be more adequate then the "parameter only" attempt. To prove this result the authors of [19] used classical methods of mathematical analysis like differentiation (up to eighth order derivatives) and taking limits. Seven years later Theorem 5.16 was generalized by Baják and Páles in the paper [8] where they solved Eq. (5.9) in full generality. To prove the next theorem they used the computer algebra system Maple Release 9 to compute the Taylor expansion of the approximation of some involved functions up to 12th order. The method of Taylor series expansion was also used by Gh. Toader, Costin and S. Toader in [154] to study the invariance problem in the class of extended logarithmic means, that is the equation which is a particular case of (5.9). Of course the final result of [154] is covered by Theorem 5. 16.
At the very end of the section we say some words about the mixed case where each of the means K, M, N satisfying the invariance equation K • (M, N ) = K is either a Gini mean, or a Stolarsky mean. There are six such equations and two of them, that is (4.13) and (5.9), have been already discussed. Then it remains to study the following four: