M-affine functions composing Sturm–Liouville families

Given an n-variable mean M defined on a real interval I, an M-affine function is a solution to the functional equation When M is a quasilinear mean, the set of continuous M-affine functions is a Sturm–Liouville family on every compact interval a,b⊆I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ a,b\right] \subseteq I$$\end{document}; i.e., for every α,β∈a,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ,\beta \in \left[ a,b\right] $$\end{document}, there exists an M-affine function f such that fa=α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left( a\right) =\alpha $$\end{document} and fb=β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f\left( b\right) =\beta $$\end{document}. The validity of the converse statement is explored in this paper and several consequences are derived from this study. New characterizations of quasilinear means and the solution to Eq. (1) under suitable conditions are among the more important ones.


Introduction and preliminaries
Let I = ∅ be a real interval. An n-variable mean M defined on I is a function M : I n → I which is internal ; i.e., it satisfies the property min{x 1 , . . . , x n } ≤ M (x 1 , . . . , x n ) ≤ max{x 1 , . . . , x n }, x 1 , . . . , x n ∈ I. (2) M is said to be strict when the inequalities (2) turn out to be strict provided that the variables x i are not all equal. Immediate consequences of (2) are both the equality M (x, ..., x) = x, x ∈ I, (which shows that means are reflexive functions) and the fact that a mean M is continuous at every point of the diagonal {(x, ..., x) : x ∈ I} of I n . A mean invariant under rearrangements of their arguments is said to be a symmetric mean, so that an n-variable mean M is symmetric when M (x σ1 , . . . , x σn ) = M (x 1 , . . . , x n ) for every σ = (σ 1 , , . . . , σ n ) ∈ S n , the symmetric group of order n. The restriction to a subinterval J ⊆ I of an n-variable mean M defined on I is an n mean on J, which will be denoted by M | J .
The set of all [continuous] n-variable means defined on an interval I will be denoted by M n (I) [CM n (I)]. When a change of variable f : I → J is performed, a given mean M ∈ CM n (I) becomes another mean N ∈ CM n (J) and, by identifying the so related means M and N , an equivalence relationship is introduced on CM n . Namely, given M ∈ CM n (I) and N ∈ CM n (J), it is said that M and N are conjugated means when there exists a homeomorphism f : I → J such that the equality where f : I → R varies on the set of strictly monotonic and continuous functions. The function f is called the generator of the quasilinear mean L f . In the literature (v.g. [8, pg. 266]; [9, pg. 215]; [14, pg. 208]), nonnegative weights are often admitted in definition (4) but, throughout this paper, quasilinear means are always strict means. (Note that the annulation of some weights in (4) simply produces a quasilinear mean in fewer variables). Particularly relevant is the equal weights (or symmetric) case: the conjugacy class of the arithmetic mean A (x 1 , x 2 , ..., x n ) = n j=1 x j /n, x 1 , x 2 , ..., x n ∈ R, is given by means of the form where, as before, f : I → R denotes a generic continuous and strictly monotonic function. These means are named quasiarithmetic means. It must be added that, in reference to means defined by (4), a non uniform terminology was employed. In the recent literature, they are frequently named weighted quasiarithmetic means, but in Chap. III of [12], the explicit denomination mean values with an arbitrary function was preferred. Given two means M ∈ CM n (I) and N ∈ CM n (J), one can look for functions f satisfying equality (3). This type of functional equations (or even a more general one in which M and N are continuous functions) have been studied since the first decades of the past century (for n = 2 see [1, pgs. 62, 79, 145], and the corresponding references; [7, pg. 239 and ff.]; [4,10,11]), but the problem of finding conditions on the means M and N in order that the functional equation (3) admits nontrivial (non constant) solutions has not been fully solved. When M = N , (3) takes the form a functional equation which can be seen as a generalization of the Jensen equation and whose solutions are, for this reason, named M -affine functions ( [10,17]). Indeed, for every n ≥ 2 and every real interval I, A-affine functions have the form f (x) = α(x) + h, where α : R → R is an additive function and b is a real constant, but continuous A-affine functions reduce to the set of affine functions Suppose that M ∈ M n (J) and ∅ = I ⊆ J. Along this paper, the general solution f : I → J to the equation will be denoted by A (M ; I, J), while the notation AC (M ; I, J) is reserved for the general continuous solution to (6). When I = J, these notations are to be simplified: The set constituted by affine functions on an interval I ⊆ R (i.e., the set AC (A; I),) will be denoted by Aff (I); i.e., ∈ R 2 such that the affine function t → mt + h is a member of Aff (I) will be denoted by AF F (I); i.e., For instance, where E ∧ denotes the convex hull of the set E. Clearly, Aff (I) and AF F (I) are convex sets regardless of the interval I. Further properties of these sets are to be considered in Sect. 5, where it will be appreciated that the visual representation AF F (I) of Aff (I) may help in clarifying some developments. This paper deals with a sort of inverse problem: deducing properties of the means M from the knowledge of some properties of A (M ; I) or AC (M ; I). For example, if for a strict mean M defined on R, the functions f (x) = mx + h, m, h ∈ R, are solutions to the equation Note that no hypothesis was made on the regularity of M . Furthermore, note that the same result is true whenever M is a strict mean defined on an interval I such that the inclusion Af f (I) ⊆ A (M ; I) holds. This fact quickly follows from the equality where x 0 , y 0 ∈ I, x 0 < y 0 . Unfortunately, this is no longer true when the number of variables is greater than 2 (cf. [1, pg. 237]): given the three-variable linear means L 1 , L 2 and L 3 , with L i = L j at least for a pair i, j, i = j, the (continuous) strict mean M (x, y, z) defined on R by M (x, x, x) = x, x ∈ R, and by when x, y ∈ R, x = y, and by when x, y, z ∈ R, x = y, y = z, z = x, serves as a counterexample. However, one can prove the following: Note that a mean fulfilling the hypotheses of the proposition is not only continuous but also differentiable at every point of the diagonal of I n .
Proof. First of all observe that, regardless of the interval I, {0}×I ⊆ AF F (I) and (m, h 0 ) ∈ AF F (I) provided that h 0 ∈ int (I) and m > 0 is small enough. Thus, if (t 0 , ..., t 0 ) ∈ (int (I)) n is the point at which M is differentiable and (which is, by the former observation, defined and continuous on an interval of the form [0, δ) (δ > 0)) has a right-hand derivative D + M at u = 0 given by Now, by the assumptions we can write and taking (right-hand) derivatives at u = 0 in this equality, Since M is a strict mean, the weights in the left hand side of equality (12) must satify where φ : I → R + is a continuous and strictly monotonic function and w = h (1).
In this paper, a class of functions F will be considered an extensive one whenever there exists a function f ∈ F passing through every pair of points.
A remarkable example of a Sturm-Liouville family is furnished by the semigroup AC (L φ ; J) corresponding to the n-variable quasilinear mean L φ with generator φ : J → R and weights w i , i = 1, . . . , n. In fact, observe that equation (6) takes, in this case, the form for an unknown function f : (13), reduces it to the equation whose general continuous solution is given by where m, h are real constants such that g(t) ∈ φ (J) , t ∈ φ (J), (this is a simple consequence of [1] , Theor. 2, pg. 67 or also [14], pg. 382 and ff.) and hence, a solution f to Eq. (13) must have the form where (m, h) ∈ AF F (φ (J)). A substitution of (15) in (13) shows that (15) really solves this equation, so that (cf. [2], Chap. 15, Prop. 6, for the case n = 2 and M symmetric) Proof. It is sufficient to observe that, for any pair of numbers α, β ∈ J, the system of equations and that the pair (m, h) given by (17) is really a member of AF F (φ ([a, b])).
the L φ -affine function passing through (a, α) and (b, β) takes the form so that, from the proof of Prop. 3 it is seen that Now, assume that M is a strict and continuous mean defined on J such that, for every compact interval [a, b] ⊆ J, the family AC (M ; [a, b]) (or even AC (L φ ; [a, b] , J)) is a Sturm-Liouville family. Must M be a quasilinear mean? This paper is addressed to answer this question. Concretely, along Sects. 2 and 3, a proof of the following result will be developed.
In other words, when for a certain strict continuous mean M , AC (M ; [a, b]) is a Sturm-Liouville family, then, there exists a unique increasing homeomor- Remarkably, when n = 2, Theorem 5 implies that M ψ is a linear mean, so that M turns out to be a quasilinear mean and then, the converse of Proposition 3 turns out to be true in this case. Now, what happens if the interval I is not compact? In Sect. 4, the following result will be shown.

Theorem 6. Let M ∈ CM 2 (I) be a two-variable mean defined on a real interval
Other consequences of Theorem 5 for two-variable means are explained in Sect. 4. Among them, a special mention is in place of the characterization of two-variable quasilinear means through the theory of bases, which is now presented by setting aside the differentiability hypothesis imposed on the means in [3]. Section 5 is devoted to studying the case of n-variable means. The following result, which can be considered as an ample generalization of Proposition 1, will be shown there.
The final Sect. 6 serves to provide some examples and remarks. In particular, the use of the above results in solving the functional equation (6) will be illustrated there.

Continuous M -affine functions constituting a Sturm-Liouville family
In the following result, whose proof can be found in [3] (see also [4] and [5]), the main properties of Aczel dyadic iterations are established.
Theorem 8. a) Let I and J be two real intervals and M ∈ M 2 (I) and N ∈ M 2 (J). If the equality which turns out to be increasing when x < y and decreasing when x > y. AEM c) For each δ ∈ (0, 1), M (δ) is a continuous strict mean defined on I (while y) is a quasiarithmetic mean with generator f , then it is inductively shown that for every δ ∈ [0, 1]. As a first application of Aczel dyadic iterations, let us prove the following: Let M ∈ CM n (J) be a continuous and strict mean defined on a real interval J and consider a compact subinterval Proof. In the first place, let us consider the case n = 2. From parts a)-b) of Theor. 8 and the continuity of f , we can write in this case whence, setting x = a, y = b, and using the notation introduced in Theor This expression and Theor.
Clearly N is a strict and continuous mean, and if f is M -affine, then it is also N -affine. Hence, the general case follows from the case n = 2. This completes the proof.
so that the continuity of α → f α,β (t) follows from Theor. 8, c). In order to prove the monotonicity, let us consider α, α ∈ [a, b] , α < α , and suppose that there and f α,β and f α ,β are both continuous functions, there exists c ∈ (a, t 0 ) such that f α,β (c) = f α ,β (c) and therefore, an argument like that used above to prove the uniqueness of f α,β shows that f α, Since t 0 > c, this is in contradiction to the former assumption and Summarizing the above discussion, the following result can be established. Proof. After the previous discussion, it remains prove only that β → f α,β (t) is monotonic and continuous on [a, b]. This is an immediate consequence of the representation and the corresponding properties of α → f α,β (t).
Under the hypotheses of Proposition 10 and remembering that AC (M ; [a, b]) is a semigroup, it turns out that, for every α i , β i ∈ [a, b] , i = 1, 2, there exists a unique pair α, β ∈ [a, b] such that and, similarly, Now, consider the function F :  (25) is a solution to the composite functional equation in the class consisting of functions with the following properties: is monotonic with respect to each variable, and strictly monotonic with respect to the variable t provided that α = β (t → F (t, α, β) is strictly increasing when α < β and strictly decreasing when α > β); iii) F (a, α, β) = α and F (b, α, β) = β.

After this result, Proposition 3 and Remark 4 imply that
where ψ : [a, b] → [0, 1] is an increasing homeomorphism, must be a solution to the functional equation (26) in the class of functions satisfying the properties i), ii) and iii). A direct checking of this fact is an easy task. As it will be seen in the next section, this expression really provides the general solution to (26).

The functional equation (26)
The purpose of this section is to prove the following: A proof of Theorem 5 will easily follow from this result. As a first observation note that, since f a,b must be an increasing homeomorphism onto [a, b] and f 2

] is a solution to the functional equation (26) satisfying conditions i), ii) and iii), then F can be written in the form
where H, G : [a, b] 2 → [a, b] are solutions to the system of functional equations which are continuous, monotonic in both variables and strictly monotonic in the first variable when α = b, while Ψ is a continuous function implicitly defined by Proof.

] is a solution to (26) satisfying conditions i), ii) and
iii), and the functions G and H are respectively defined by and then both G and H turn out to be continuous on [a, b] 2 by condition i), while condition ii) shows that G and H must be monotonic functions in both variables. Moreover, t → G (t, α) is strictly monotonic for every α = b and the H (t, b)). For this reason, the (continuous) function Ψ defined by is the unique solution to Eq. (30). Now, let us prove that system (29) is solved by the above defined functions G and H . In fact, from (26) and condition iii) it is deduced that F (α, β, b) Analogously, it can be written that It remains to prove that, in terms of G and H, F is expressed by (28). To this end, first consider the case α ≤ β; thus, from (26) and condition iii), it is derived that whence, introducing β 1 = Ψ (β, α) and taking (30) into account, we obtain Similarly, when α > β, it can be written that and the substitution α 1 = Ψ (α, β) gives This completes the proof. Remark 14. Note that the function G is really increasing in both variables and strictly increasing in the first variable when α = b. In its turn, H is strictly decreasing in the first variable when α = b, while it is increasing in the second variable.
In the next paragraph, the system of composite Eq. (29) is to be solved. The first equation in this system is no other than the associativity equation. Fortunately, its solution in our setting is furnished by a result due to C. H. Ling (see [15], Main Theorem, or also [16] Theor. 3.2). In the next paragraphs, R and [0, +∞] will stand respectively for the sets of extended real numbers and nonnegative extended real numbers.
The function G in Proposition 13 is easily shown to satisfy the hypotheses of Ling's theorem. Moreover, from the strict monotonicity of G in the first variable it follows that f (b) = +∞. In fact, the assumption In order to solve the second equation in (29), let us substitute the expression (31) for G in it to obtain Setting ξ = f (t) , η = f (α) , ζ = f (β) and introducing the function K :

this equation can be written as
or, equivalently,

L. R. Berrone and G. E. Sbérgamo AEM
In other words, the function (ξ, η) → K (ξ, η) − η depends only on ξ; i.e., there exists p : and therefore The replacement α = a in the last expression produces whence it is deduced that p is a strictly decreasing involutory function satisfying p (0 + ) = +∞ and p (+∞) = 0. An expression for the function Ψ of Proposition 13 is promptly derived from (32) in the form From the above discussion and Proposition 13, it follows that any solution to Eq. (26) satisfying conditions i), ii) and iii) can be written as where f : I → [0, +∞] is a continuous and strictly increasing function with f (a) = 0 and p is a strictly decreasing involutory function satisfying p (0 + ) = +∞ and p (+∞) = 0. Now, assume that the function F represented by (33) is a solution to Eq. (26); then, taking α 1 , β 1 , β 2 ∈ [a, b] with α 1 ≤ β 1 , it can be written that and, in view of f (a) = 0, it follows that where s = f (t) , a 1 = f (α 1 ) and b i = f (β i ) , i = 1, 2. On the other hand, whence, since F (α 1 , α 2 , β 2 ) ≤ F (β 1 , α 2 , β 2 ), the following equality is deduced where again s = f (t) , a 1 = f (α 1 ) and b i = f (β i ) , i = 1, 2. Since the left hand sides of (34) and (35) are equal, their corresponding right hand sides must be equal as well and therefore, the equality must hold for every s, a 1 , b 1 , b 2 ∈ [a, b] or, after the substitutions x = b 1 − a 1 , y = a 1 , z = p (b 2 ) and s = p (s), where x, y, z, s ∈ R + 0 . Summarizing the above developments, the following result can be established. Proof. The proof follows from Proposition 13 and the preceding discussion.
In this way, the substitutions s = p (s) and z = p (z) in (37) enables us to write it in the form Note that 0 ≤ z − yΔz ≤ z, with the inequalities strict provided that y, z > 0. As it is shown by the following result, the function s → sΔz has nice properties. [0, +∞] be the quasisum defined by (38). Then, for every z ∈ R + , the function s → sΔz is strictly subadditive, strictly increasing, strictly concave and continuously differentiable in R + .
By commutativity, the function z → sΔz has the same properties as s → sΔz.
Proof. Fix z ∈ R + and consider the function s → sΔz. Since p is a strictly decreasing function, s → sΔz turns out to be strictly increasing. As a consequence, (39) and the inequality z − tΔz < z yields (s + t) Δz = sΔ (z − tΔz) + tΔz < sΔz + tΔz, s, t ∈ R + ; i.e., s → sΔz is subadditive. To prove the strict concavity of s → sΔz, choose a pair s, t ∈ R + with s = t, say s < t; then, a repeated use of (39) produces where the last inequality holds by the strict monotonicity of z → sΔz. On the other hand, so that, combining (40) and (41) we deduce or, equivalently, By symmetry, inequality (42) holds also when s > t and, due to the continuity of s → sΔz, it implies the strict concavity of this function. Now, for every s ∈ R + , the existence of the lateral derivatives D + s (sΔz) and D − s (sΔz) is ensured by the concavity of s → sΔz. In particular, in view of (39), for the right derivative D + s (sΔz) it can be written that The last of these equalities was obtained by replacing t = p (u) . Since 0 ≤ z − s z ≤ z and z ∈ R + was arbitrarily chosen, it is concluded that the function is defined for every λ ≥ 0. Clearly, Φ is decreasing and the equalities hold for every λ, μ ≥ 0. In other words, Φ is a decreasing solution to the exponential Cauchy equation and, in consequence, Φ (λ) ≡ 0 or Φ (λ) ≡ e −kλ for any k ≥ 0. Indeed, the instances Φ = 0 or Φ = 1 must be excluded since, in these cases, we would have D + s (sΔz) ≡ 0 or D + s (sΔz) ≡ 1 and therefore, D s (sΔz) ≡ 0 or D s (sΔz) ≡ 1, two identities contradicting the strict concavity of s → sΔz. In this way, there exists k > 0 such that This equality shows that s → D + s (sΔz) is continuous on R + and hence, there exists the standard derivative D s (sΔz) and This completes the proof. Proof. Let us denote by Diff (p) the set of points where the derivative p exists. By Lebesgue´s Theorem, Diff (p) contains almost every point of R + so that, for a given s ∈ R + , one can chose t, z 0 > 0 such that p (s) + t and p (p (s) + t) + p (z 0 ) are both in Diff (p). Thus, the chain rule applied to s → sΔz 0 = p (p (s) + p (z 0 )) at p (s) + t yields Now, in view of (43), so that we must have p (p (s) + t) = 0, and therefore This shows that s ∈ Diff (p), and thus Diff (p) = R + . Now, from (38) and (43) it is obtained that p (p (s) + p (z)) p (s) = e −kp(z−p(p(s)+p(z))) , s, z ∈ R + ; or, replacing s = p (s) and z = p (z), whence the continuity of p on (s, +∞) is easily derived. Since s can be arbitrarily chosen in R + , p turns out to be continuous on R + . Moreover, making z ↑ +∞ in (44) yields Finally, p being an involutory function, it turns out that whence, in view of (45) and the fact that p is strictly decreasing, we deduce This completes the proof.
Now, a new application of (46) produces lim x↑+∞ p (p (x)) p (p (x + y)) − p (p (y)) = lim where the last equality follows from the fact that p (+∞) = 0. Moreover, from (46) and (44) with s = x and z = y, it is obtained that Now, from (49), (50) and (51) we deduce where s, y ∈ R + and k > 0 is a constant. Substituting s = p (s) in the last member of these equalities, gives whence, for every s, y ∈ R + , 1 p (s + y) = p (p (s + y)) = p (p (y)) + p (p (s)) e ky = 1 p (y) The first member of these equalities is symmetric in its arguments, which shows that 1 p (y) + 1 p (s) e ky = 1 p (s) + 1 p (y) e ks or, equivalently, In other words, there exist a positive constant A such that An integration of the equality (52) yields which holds if and only if A = 1. This proves that a solution to Eq. (36) which satisfies the hypotheses of the proposition must be of the form (47). A simple substitution shows that (47) is really a solution to Eq. (36). This completes the proof.