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 $$\left[ a,b\right] \subseteq I$$a,b⊆I; i.e., for every $$\alpha ,\beta \in \left[ a,b\right] $$α,β∈a,b, there exists an M-affine function f such that $$f\left( a\right) =\alpha $$fa=α and $$ f\left( b\right) =\beta $$fb=β. 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 6 = ; be a real interval. A n variables mean M de…ned on I is a function M : I n ! I which is internal ; i.e., it satis…es the property minfx 1 ; : : : ; x n g M (x 1 ; : : : ; x n ) maxfx 1 ; : : : ; x n g; x 1 ; : : : ; x n 2 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 the (2) are both the equality M (x; :::; x) = x; x 2 I; (which show that means are re ‡exive functions) and the fact that a mean M is continuous at every point of the diagonal f(x; :::; x) : x 2 Ig of I n . A mean invariant under rearrangements of their arguments is said to be a symmetric mean, so that a n variables mean M is symmetric when M (x 1 ; : : : ; x n ) = M (x 1 ; : : : ; x n ) for every = ( 1 ; ; : : : ; n ) 2 S n , the symmetric group of order n. The restriction to a subinterval J I of a n variables mean M de…ned on I is a n mean on J which will be denoted by M j J .
The set of all [continuous] n variables means de…ned 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 2 CM n (I) becomes another mean N 2 CM n (J) and, by identifying the so related means M and N , an equivalence relationship is introduced on CM n . Namely, given M 2 CM n (I) and N 2 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 f (M (x 1 ; x 2 ; :::; x n )) = N (f (x 1 ) ; f (x 2 ) ; :::; f (x n )) ; ( holds for every x 1 ; x 2 ; :::; x n 2 I. This relationship decomposes CM n (I) in classes named conjugacy classes. For instance, the conjugacy class of the linear mean L (x 1 ; :::; x n ; w) = n X j=1 w j x j ; (where the coe¢ cients w i , the weights of the mean, satisfy w i > 0; i = 1; : : : ; n; 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 de…nition (4) but, throughout this paper, quasilinear means are ever 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 ) = P n j=1 x j =n; x 1 ; x 2 ; :::; x n 2 R, is given by the 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 the means de…ned 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.
A well known result (cf. [12], Sect. 3.2; [1], Theor. 2, pg. 67; [2], Cor. 5, pg. 246; [14], pg. 382 and ¤.) establishes that the generator f of a quasilinear mean M de…ned on I is determined only up to an a¢ ne homeomorphism by M : the equality L f = L g holds if and only if g = mf + h for certain real constants m and h, m 6 = 0. On the other hand, the di¤erentiability of the quasilinear mean L f is strictly related to the di¤erentiability of its generator, as established by the following: Proposition 1 A quasilinear mean L f de…ned on I is di¤ erentiable if and only if its generator f is di¤ erentiable in I.
Proof. The "if" part easily follows from the chain rule. To prove the converse it is enough to consider two variables means M (x; y), for which (4) can be rewritten in the form f (x) = 1 w 1 (f (M (x; y)) w 2 f (y)) ; x; y 2 I.
Since f is di¤erentiable almost everywhere in I by the Lebesgue´s Theorem, for a given x 2 I, there exists y 2 I such that f is di¤erentiable at the point M (x; y). This fact and the chain rule applied to the right hand side of (6) show that f is di¤erentiable at x. The proposition follows from the arbitrariness of x 2 I. Given two means M 2 CM n (I) and N 2 CM n (J), one can look for functions f satisfying the 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 …rst decades of the past century (for n = 2 see [1], pgs. 62, 79, 145, and the corresponding references; [7], pg. 239 and ¤.; [10]; [4], [11]), but the problem of …nding conditions on the means M and N in order that functional equation (3) admits nontrivial (non constant) solutions has not been fully solved. When M = N , (3) takes the form f (M (x 1 ; :::; x n )) = M (f (x 1 ) ; :::; f (x n )) ; x 1 ; :::; x n 2 I; a functional equation which can be seen as a generalization of the Jensen equation ; x 1 ; :::; x n 2 I; (8) and whose solutions are, by this reason, named M -a¢ ne functions ( [10], [17]). Indeed, for every n 2 and every real interval I, the A-a¢ ne functions have the form f (x) = (x) + h, where : R ! R is an additive function and b is a real constant, but the continuous A-a¢ ne functions reduce to the set of a¢ ne functions f (x) = mx + h; m; h 2 R. Along this paper, the general solution to (7); i.e., the set of M -a¢ ne functions, will be denoted by A (M ; I), while the notation AC (M ; I) is reserved for the continuous M -a¢ ne functions. Clearly, A (M ; I) and AC (M ; I) are semigroups under " ", the usual composition of functions.
The set constituted by the a¢ ne functions on an interval I R (i.e., the set AC (A; I),) will be denoted by A (I); i.e., The set of parameters (m; h) 2 R 2 such that the a¢ ne function t 7 ! mt+h is a member of A (I) will be denoted by AF F (I); i.e., AF F (I) = (m; h) 2 R 2 : mI + h I : In this way, AF F (R + ) = R + 0 R + and AF F ([0; 1]) = f(0; 0) ; (1; 0) ; (0; 1) ; ( 1; 1)g^, where E^denotes the convex hull of the set E. Clearly, A (I) and AF F (I) are convex sets whichever be the interval I. Further properties of these sets are to be considered in Section 5, where it will be appretiated that the visual representation AF F (I) of A (I) can clarify some developments.
This paper deals with a sort of inverse problem: to deduce 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 de…ned on R, the functions f (x) = mx + h; m; h 2 R, were solutions to the equation or, in other terms, if the inclusion Af f (R) A (M ; I) holds, then necessarily M (x; y) = M (0 (y x) + x; 1 (y x) + x) = M (0; 1) (y x) + x is a linear mean (cf. [1], Theor. 1, pg. 234). Note that no hypothesis was made on the regularity of M ; furthermore, note that the same result is true whenever M is strict mean de…ned on an interval I provided that the inclusion Af f (I) A (M ; I) holds, a fact that quickly follows from the equality where x 0 ; y 0 2 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 variables linear means L 1 ; L 2 and L 3 , with L i 6 = L j at least for a pair i; j, i 6 = j, the (continuous) strict mean M (x; y; z) de…ned on R by M (x; x; x) = x; x 2 R, by when x; y 2 R, x 6 = y, and by when x; y; z 2 R, x 6 = y; y 6 = z; z 6 = x, serves as a counterexample. However, it can be proved the following: Proposition 2 Let I be a real interval with int (I) 6 = ; and M 2 CM n (I) be a strict mean such that Af f (I) A (M ; I). If M is di¤ erentiable at a point of the diagonal of (int (I)) n , then M is a linear mean.
Note that a mean ful…lling the hypotheses of the proposition is not only continuous but also di¤erentiable at every point of the diagonal of I n .
In this paper, a class of functions F will be considered an extense one whenever there exists a function A remarkable example of a Sturm-Liouville family is furnished by the semigroup AC (L ; J) corresponding to the n variables quasilinear mean L with generator : J ! R and weights w i ; i = 1; : : : ; n. In fact, observe that equation (7) takes, in this case, the form ! ; x i 2 J; i = 1; : : : ; n; (15) for a unknown function f : J ! J. Replacing g = f 1 : (J) ! (J) in (??), reduce it to the equation ; t i 2 (J) ; i = 1; : : : ; n; whose general continuous solution is given by where m; h are real constants such that g(t) 2 (J) ; t 2 (J), (this is a simple consequence of [1], Theor. 2, pg. 67 or also [14], pg. 382 and ¤.) and hence, a solution f to equation (15) must have the form where (m; h) 2 AF F ( (J)). A replacement of (17) in (??) shows that (17) really solves this equation, so that (cf. [2], Chap. 15, Prop. 6, for the case n = 2 and M symmetric) A straightforward consequence of this characterization of AC (L ; J) is the following: Proof. It is su¢ cient to observe that, for any pair of numbers ; 2 J, the system of equations has the solution and that the pair (m; h) given by (18) really is a member of AF F ( ([a; b])).
it turns out to be Now, assume that M is a strict and continuous mean de…ned on J such that, for every compact interval [a; b] J, the family AC M j [a;b] ; [a; b] is a Sturm-Liouville family. ¿Must be M a quasilinear mean? This paper is addressed to answer to this question. Concretely, along Sections 2 and 3, a proof of the following result will be developed.
where (m; h) 2 AF F ([0; 1]). Remarkably, when n = 2, Theor. 6 implies that M is a linear mean, so that M turns out to be a quasilinear mean, so that the converse of Prop. 4 turns out to be true in this case. Now, ¿what if the interval I is not compact? In Section 4, the following result will be shown. Other consequences of Theor. 6 for two variables means are explained in Section 4. Among them, a special mention deserves the characterization of two variables quasilinear means through the theory of bases, which is now presented by setting aside the technical di¤erentiability hypotheses made in [3]. Section 5 is devoted to study the case of n variables means. The following result, which can be considered as an ample generalization of Prop. 2, will be shown there.
Theorem 8 Let M 2 CM n (I) be a n variables, strict and continuous mean de…ned on a real interval I. If f[a k ; b k ] : k 2 Ng is a sequence of nested and exhaustive compact subintervals of I such that AC (M ; [a k ; b k ]) is a Sturm-Liouville family for every k 2 N, then there exists a strictly increasing and continuous function : I ! R such that AC (M ; (I)) = Af f ( (I)). Furthermore, M is a quasilinear mean in I provided that M is di¤ erentiable.
The …nal Section 6 serves to gather together some examples and remarks. Particularly, the use of the above results in solving the functional equation (7) will be illustrated there. is de…ned on I 2 as follows: …x x; y 2 I and set then, assuming that M j 2 n (x; y) is known for n 0 and every 0 j 2 n , de…ne In the following result, whose proof can be found in [3] (see also [4] and [5]), the main properties of the Aczel dyadic iterations are established.
As a …rst application of the Aczel dyadic iterations, let us prove the following: Proof. In the …rst place, let us consider the case n = 2: From part a) of Theor. 9 and the continuity of f , it can be written in this case whence, setting x = a, y = b, and using the notation introduced in Theor. 9 This expression and Theor so that the continuity of 7 ! f ; (t) follows from Theor. 9, c). In order to prove the monotonicity, let us consider ; 0 2 [a; b] ; < 0 , and suppose that there (a) and of f ; and f 0 ; are both continuous functions, there exists c 2 (a; t 0 ) such that f ; (c) = f 0 ; (c) and therefore, an argument like that used above to prove the uniqueness of f ; shows that f Since t 0 > c, this is in contradiction with the former assumption and Summarizing the above discussion, the following result can be established. Proof. After the previous discussion, it remains prove only that 7 ! f ; (t) is monotonic and continuous on [a; b]. This is a immediate consequence of the representation and the corresponding properties of 7 ! f ; (t).
Under the hypotheses of Prop. 11 and remembering that AC M j [a;b] ; [a; b] is a semigroup, it turns out to be that, for every i ; i 2 [a; b] ; i = 1; 2, there exist a unique pair ; 2 [a; b] such that and, similarly, for every i ; i 2 [a; b] ; i = 1; 2, it can be written Now, consider the function F : (26) in the class constituted by the functions with the following properties: ) is monotonic with respect to each variable, and strictly monotonic with respect to the variable t provided that 6 = (t 7 ! F (t; ; ) is strictly increasing when < and strictly decreasing when > ); iii) F (a; ; ) = and F (b; ; ) = .
Proof. The functional equation (26) is a rewriting of (24) using (25). ii) and iii) are an immediate consequence of (25) and Props. 10 and 11 (the strict monotonicity of t 7 ! F (t; ; ) follows from the representation (22) and Theor. 9). In regards to i), observe that the function F : is separately continuous and monotonic in each variable and therefore, F is continuous. In fact, the argument employed by R. L. Kruse and J. J. Deely in [13], Prop. 2, to prove the joint continuity on an given open set can be easily extended to prove joint continuity on the cube [a; b] 3 . After this result, Prop. 4 and Remark 5 imply that 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: The general solution to the functional equation (26) in the class of functions ful…lling the conditions i), ii) and iii) is given by is the unique increasing homeomorphism satisfying (27).
A proof of Theor. 6 will easily follow from this result. As a …rst observation note that, in view of f a;b must be an increasing homeomorphism onto [a; b] and f 2 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 7 ! G (t; ) is strictly monotonic for every 6 = b and the same is true for t 7 ! H (t; ) (but G (t; b) b H (t; b)). By this reason, the (continuous) function de…ned by is the unique solution to equation (30). Now, let us prove that system (29a) is solved by the above de…ned functions G and H. In fact, from (26) and condition iii) it is deduced It remains to prove that, in terms of G and H, F is expressed by (28). To this end, …rst consider the case ; thus, from (26) and condition iii), it is derived This completes the proof.

Remark 15
Note that the function G is really increasing in both variables and strictly increasing in the …rst variable when 6 = b. In its turn, H is strictly decreasing in the …rst variable when 6 = b, while it is increasing in the second variable.
In the next paragraph, the system of composite equations (29a) is to be solved. The …rst 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 ). In the next paragraphs, R and [0; +1] will stand respectively for the extended real numbers and the nonnegative extended real numbers. The function G in Prop. 14 is easily cheeked to satisfy the hypotheses of Ling's theorem. Moreover, from the strict monotonicity of G in the …rst variable it follows that In order to solve the second equation in (29a), let us substitute the expression (31) for G in it to obtain In other words, the function ( ; ) ! K ( ; ) depends only on ; i.e., there exists p : and therefore H (t; ) = f 1 (p (f (t)) + f ( )) : The replacement = a in the last expression produces whence it is deduced that p is a strictly decreasing involutory function satisfying p (0 + ) = +1 and p (+1) = 0. An expression for the function of Prop. 14 is promptly derived from (32) in the form From the above discussion and Prop. 14, it follows that any solution to equation (26) satisfying conditions i), ii) and iii) can be written as where f : I ! [0; +1] is a continuous and strictly increasing function with f (a) = 0 and p is a strictly decreasing involutory function satisfying p (0 + ) = +1 and p (+1) = 0. Now, assume that the function F represented by (33) is a solution to equation (26); then, taking 1 ; 1 ; 2 2 [a; b] with 1 1 , it can be written and, in view of f (a) = 0, it follows that On the other hand, whence, since F ( 1 ; 2 ; 2 ) F ( 1 ; 2 ; 2 ), the following equality is deduced F (t; F ( 1 ; 2 ; 2 ) ; F ( 1 ; 2 ; 2 )) = f 1 (p (p (s) + p (p (p (b 1 ) + p (b 2 )) p (p (a 1 ) + p (b 2 )))) +p (p (a 1 ) + p (b 2 ))) ; where again s = f (t) ; a 1 = f ( 1 ) and b i = f ( 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 p (p (p (p (s) + p (b 1 a 1 )) + a 1 ) + p (b 2 )) = p (p (s) + p (p (p (b 1 ) + p (b 2 )) p (p (a 1 ) + p (b 2 )))) + p (p (a 1 ) + p (b 2 )) ; must hold for every s; a 1 ; b 1 ; b 2 2 [a; b] or, after the substitutions x = b 1 a 1 ; y = a 1 ; z = p (b 2 ) and s = p (s), p (p (p (s + p (x)) + y) + z) = p (s + p (p (p (x + y) + z) p (p (y) + z)))+p (p (y) + z) (36) where x; y; z; s 2 R + 0 . Summarizing the above developments, the following result can be established. Proof. The proof follows from Prop. 14 and the preceding discussion. Now, let us pay attention to the functional equation (36). In the …rst place, observe that in view of the continuity of p and the the fact that p (+1) = 0, taking limits when x " +1 in (36) produces p (p (p (s) + y) + z) = p (s + p (p (z) p (p (y) + z))) + p (p (y) + z) ; (37) where x; y; z; s 2 R + 0 . In order to simplify the expressions, let us de…ne a commutative operation (quasisum) : [0; +1] 2 ! [0; +1] by x y = p (p (x) + p (y)) : In this way, the substitutions s = p (s) and z = p (z) in (37) enables to write it in the form (s + y) z = s (z y z) + y z: Note that 0 z y z z, being the inequalities strict provided that y; z > 0. As it is shown by the following result, the function s 7 ! s z has nice properties. : [0; +1] 2 ! [0; +1] be the quasisum de…ned by (38). Then, for every z 2 R + , the function s 7 ! s z is strictly subadditive., strictly increasing, strictly concave and continuously di¤ erentiable in R + .
By commutativity, the function z 7 ! s z has the same properties as s 7 ! s z. Proof. Fix z 2 R + and consider the function s 7 ! s z. Since p is a strictly decreasing function, s 7 ! s z turns out to be strictly increasing. As a consequence, (39) and the inequality z t z < z yields (s + t) z = s (z t z) + t z < s z + t z; s; t 2 R + ; i.e., s 7 ! s z is subadditive. To prove the strict concavity of s 7 ! s z, choose a pair s; t 2 R + with s 6 = t, say s < t; then, a repeated use of (39) produces where the last inequality holds by the strict monotonicity of z 7 ! s z. On the other hand, so that, combining (40) and (41) it is deduced or, equivalently, s + t 2 z > s z + t z 2 : By symmetry, inequality (42) holds also when s > t and, due to the continuity of s 7 ! s z, it implies the strict concavity of this function. Now, for every s 2 R + , the existence of the lateral derivatives D + s (s z) and D s (s z) is ensured by the concavity of s 7 ! s z. In particular, in view of (39), for the right derivative D + s (s z) it can be written The last of these equalities was obtained by replacing t = p (u). Since 0 z s4z z and z 2 R + was arbitrarily chosen, it is concluded that the function is de…ned for every 0. Clearly, is decreasing and the equalities hold for every ; 0. In other words, is a decreasing solution to the exponential Cauchy equation and, in consequence, ( ) 0 or ( ) e k for any k 0. Indeed, the instances = 0 or = 1 must be excluded since, in these cases, it would be D + s (s z) 0 or D + s (s z) 1 and therefore, D s (s z) 0 or D s (s z) 1, two identities contradicting the strict concavity of s 7 ! s z. In this way, there exists k > 0 such that This equality shows that s 7 ! D + s (s z) is continuous on R + and hence, there exists the standard derivative D s (s z) and Proof. Let us denote by Di (p) the set of points where the derivative p 0 exists. By the Lebesgues´s Theorem, Di (p) contains almost every point of R + so that, for a given s 2 R + , one can chose t; z 0 > 0 such that p (s) + t and p (p (s) + t) + p (z 0 ) are both in Di (p). Thus, the chain rule applied to s 7 ! s z 0 = p (p (s) + p (z 0 )) at p (s) + t yields D s (s z 0 )j s=p(s)+t = p 0 (p (p (s) + t) + p (z 0 )) p 0 (p (s) + t) : Now, in view of (43), D s (s z 0 )j s=p(s)+t = e kp(z0 (p(s)+t)4z0) > 0; so that it must be p 0 (p (s) + t) 6 = 0, and therefore This shows that s 2 Di (p), and thus Di (p) = R + . Now, from (38) and (43) it is obtained p 0 (p (s) + p (z)) p 0 (s) = e kp(z p(p(s)+p(z))) ; s; z 2 R + ; or, replacing s = p (s) and z = p (z), whence the continuity of p 0 on (s; +1) is easily derived. Since s can be arbitrarily chosen in R + , p 0 turns out to be continuous on R + . Moreover, making z " +1 in (44) yields Finally, being p an involutory function, it turns out to be p 0 (p (s)) p 0 (s) = 1; s 2 R + ; whence, in view of (45) and the fact that p is strictly decreasing, it is deduced This completes the proof. It should be noted that an inductive reasoning based on (44) shows that p is really a C 1 function in R + . At this point, the solutions to equation (36) can be determined. Proof. By Prop. 19, p is continuously di¤erentiable in R + . Thus, deriving both members of (36) with respect to z and then taking limits when z # 0, it is obtained p 0 (p (p (s + p (x)) + y)) = p 0 (s + p (x)) p 0 (x) (p 0 (p (x + y)) p 0 (p (y)))+p 0 (p (y)) : Observe that p 0 (p (x + y)) p 0 (p (y)) 6 = 0 for every x; y 2 R + . In fact, if p 0 (p (x + y)) = p 0 (p (y)) for any pair x; y 2 R + ; then, p 0 (x + y) = p 0 (y) by (46), an equality which, together (44) with s = y and z = x, would imply 1 = p 0 (y) p 0 (p (y)) = p 0 (x + y) p 0 (p (y)) = e kp(p(x) p(x+y)) ; whence p (p (x) p (x + y)) = 0: Since p (x) p (x + y) 2 R + , the last equality is an absurdity. In this way, (48) can be rewritten in the form p 0 (x) = p 0 (p (p (s + p (x)) + y)) p 0 (p (y)) p 0 (s + p (x)) (p 0 (p (x + y)) p 0 (p (y))) : and then, using (46), it is deduced = lim x"+1 p 0 (p (p (s + p (x)) + y)) p 0 (p (y)) p 0 (s + p (x)) (p 0 (p (x + y)) p 0 (p (y))) p 0 (p (x)) = lim x"+1 p 0 (p (p (s + p (x)) + y)) p 0 (p (y)) p 0 (s + p (x)) lim x"+1 p 0 (p (x)) p 0 (p (x + y)) p 0 (p (y)) = p 0 (p (p (s) + y)) p 0 (p (y)) p 0 (s) lim x"+1 p 0 (p (x)) p 0 (p (x + y)) p 0 (p (y)) ; whence, for every s; y 2 R + , lim x"+1 p 0 (p (x)) p 0 (p (x + y)) p 0 (p (y)) = p 0 (s) p 0 (p (p (s) + y)) p 0 (p (y)) : Now, a new application of (46) produces where s; y 2 R + and k > 0 is a constant. Substituting s = p (s) in the last member of these equalities, gives e ky = p 0 (p (s)) p 0 (p (s + y)) p 0 (p (y)) ; whence, for every s; y 2 R + , = p 0 (p (s + y)) = p 0 (p (y)) + p 0 (p (s)) e ky = 1 p 0 (y) The …rst member of these equalities is symmetric in its arguments, which shows that

Two variables means
The following result, which is not devoid of intrinsic interest, will be the key to derive Theor. 7 from Theor. 6. Proof. The hypotheses ensure the existence, for every k 2 N, of a strictly monotonic and continuous function k : [a k ; b k ] ! R and a real number w k 2 (0; 1) such that Since the second member of (56) is not altered by taking k instead of k , it can be assumed that k is strictly increasing. Now, in view of M j In this way ([1], Theor. 2, pg. 67 or also [14], pg. 382 and ¤.), w k = w j ; k; j 2 N, and, for certain p k;j ; q k;j 2 R, p k;j 6 = 0, k;j (t) = p k;j t + q k;j ; t 2 [a l ; b l ] ; hence Note on one hand that, in view of k and j are both strictly increasing functions, it must really occur that p k;j > 0 for every k; j 2 N, and, on the other, that the equality (56) can be written in the form In what follows, a particular instance of (57) will be used; namely, setting p k = p k+1;k and q k = q k+1;k for every k 1, (57) takes the form Now, de…ne a sequence of strictly increasing and continuous functions k : [a k ; b k ] ! R; k 2 N, by 1 (t) = 1 (t) ; t 2 [a 1 ; b 1 ], and for k 1, From (59) it is deduced that, for every t 2 [a k ; b k ], so that the expression The same reasoning applied to the interval [a k ; b k ] yields, for every k 2 N, where k : [a k ; b k ] ! [0; 1] is an increasing homeomorphism and w k 2 (0; 1). This proves that, for every k 2 N, the restriction M j [a k ;b k ] to the subinterval [a k ; b k ] is a quasilinear mean. By Prop. 22, this implies that M is quasilinear on I.  (21) it is obtained and therefore, the base mean does not depend on P 2 I 2 .
A slightly more involved computation shows that a quasilinear mean L f possesses a unitary base as well. Now, ¿what can be said on a two variables, strict an continuous mean M when its base is a unitary set? In [3] the following result was established.
Theorem 25 Let M 2 CM 2 (I) be a di¤ erentiable strict mean. Then, the base mean of M is a unitary family if and only M is a quasilinear mean.
Let us see that, with the help of Theor. 7, the di¤erentiability hypothesis in the above statement can be omitted. Proof. The "if"part of the proof proceeds along the same lines of the particular case in which M is quasiarithmetic. The details can be seen in [3]. To prove the converse, suppose that B(M ) = f g is a base of M ; then, given a; b 2 I, with where T : R 2 ! R 2 is the transformation given by The following result will play a relevant role in the proof of Theor. 8.
or, since S is closed in AF F ([0; 1)), The closure operator in the above equalities is taken with respect to relative topology induced on AF F ([0; 1)) by the usual topology of R 2 . shows that (m; h) 2 S. It has been thus proved that S = R 2 = AF F (R).
Proof of Theor. 8. Let M 2 CM n (I) be a continuous and strict mean de…ned on I. Suppose that f[a k ; b k ] : k 2 Ng is a nested and exhaustive sequence of compact subintervals of I, and that AC ( M j ; [a k ; b k ]) is a Sturm-Liouville family for every k 2 N. Clearly, the two variables mean N de…ned on I by (23) satis…es the hypotheses of Teor. 7, so that there exist both a strictly monotonic and continuous function : I ! R and a number w 2 (0; 1) such that N (x; y) = 1 ((1 w) (x) + w (y)) ; x; y 2 I. follows from a simple specialization of the variables in the equation (7). Let us see that the above inclusion is really an equality. With this purpose, let us note that which shows that (m; h) 2 S. Since the constants are contained S, all the hypotheses of Prop. 27 are ful…lled by S, and therefore S = Af f ( (I)), as a¢ rmed. Finally, assuming that the mean M is di¤erentiable, the mean N given by (68) turns out to be di¤erentiable as well, and then, the generator is di¤erentiable in I by Prop. 1. In consequence, the conjugated mean M is di¤erentiable and AC (M ; (I)) = Af f ( (I)). By Prop. 2, M turns out to be a linear mean on (I), and therefore, M is quasilinear on I. This completes the proof.
Before …nishing this section, a result is proved which will show its usefulness in the next one.
Example 30 Consider a strict and continuous mean M de…ned on R for which the inclusion Af f (R) AC (M ; R) is satis…ed. The mean M de…ned by (10) and (10)  as a¢ rmed.