On the equality of two-variable general functional means

Given two functions $f,g:I\to\mathbf{R}$ and a probability measure $\mu$ on the Borel subsets of $[0,1]$, the two-variable mean $M_{f,g;\mu}:I^2\to I$ is defined by $$ M_{f,g;\mu}(x,y) :=\bigg(\frac{f}{g}\bigg)^{-1}\left( \frac{\int_0^1 f\big(tx+(1-t)y\big)d\mu(t)} {\int_0^1 g\big(tx+(1-t)y\big)d\mu(t)}\right) \qquad(x,y\in I). $$ This class of means includes quasiarithmetic as well as Cauchy and Bajraktarevi\'c means. The aim of this paper is, for a fixed probability measure $\mu$, to study their equality problem, i.e., to characterize those pairs of functions $(f,g)$ and $(F,G)$ such that $$ M_{f,g;\mu}(x,y)=M_{F,G;\mu}(x,y) \qquad(x,y\in I) $$ holds. Under at most sixth-order differentiability assumptions for the unknown functions $f,g$ and $F,G$, we obtain several necessary conditions for the solutions of the above functional equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarevi\'c means and of Cauchy means.


Introduction
Throughout this paper I will stand for a nonempty open real interval. In the sequel, the classes of continuous strictly monotone and continuous positive real-valued functions defined on I will be denoted by CM(I) and CP(I), respectively.
In general, a continuous function M : I 2 → I is called a two-variable mean on I if the so-called mean value inequality min(x, y) ≤ M (x, y) ≤ max(x, y) (x, y ∈ I) holds. If, for x = y, both of the inequalities in (1) are strict, then M is called a two-variable strict mean. The arithmetic and geometric means are well known instances for strict means on R + . Given a function ϕ ∈ CM(I), the two-variable quasiarithmetic mean generated by ϕ is the function A ϕ : I 2 → I defined by A ϕ (x, y) := ϕ −1 ϕ(x) + ϕ(y) 2 (x, y ∈ I).
The systematic treatment of these means was first given by Hardy, Littlewood and Pólya [10]. The most basic problem, the characterization of the equality of these means, states that A ϕ and A ψ are equal to each other if and only if there exist two real constants a = 0 and b such that ψ = aϕ + b. The characterization of quasiarithmetic means was solved independently by Kolmogorov [13], Nagumo [23], de Finetti [9] for the case when the number of variables is non-fixed. For the two-variable case, Aczél [1], [2], [3], [4] (seel also [5]), proved a characterization theorem involving the notion of bisymmetry. This result was extended to the n-variable case by Maksa-Münnich-Mokken [22].
In this paper, we consider the following generalization of quasiarithmetic means, which was introduced in [19] and also investigated in [20]. Given two continuous functions f, g : I → R with g ∈ CP(I), f /g ∈ CM(I) and a probability measure µ on the Borel subsets of [0, 1], the two-variable mean M f,g;µ : I 2 → I is defined by M f,g;µ (x, y) := f g −1 1 0 f tx + (1 − t)y dµ(t) 1 0 g tx + (1 − t)y dµ(t) (x, y ∈ I). (2) Means of the above form, will be called generalized quasiarithmetic means.
The first important particular case of this definition is when µ = δ 0 +δ 1 2 . Here and in the sequel, δ τ will denote the Dirac measure concentrated at the point τ ∈ [0, 1]. If ϕ ∈ CM(I), and p ∈ CP(I), then M ϕ·p,p;µ = B ϕ,p , which is defined by B ϕ,p (x, y) := ϕ −1 p(x)ϕ(x) + p(y)ϕ(y) p(x) + p(y) (x, y ∈ I) and is called the two-variable Bajraktarević mean (cf. Bajraktarević [6], [7]). By taking p = 1, we can see that this class of means includes quasiarithmetic means. Assuming 6 times continuous differentiability, the equality problem of these means was solved by Losonczi [14], [18]. The second important particular case is when µ is the Lebesgue measure on [0, 1] and ϕ, ψ : I → I are continuously differentiable functions with ψ ′ ∈ CP(I) and ϕ ′ /ψ ′ ∈ CM(I). Then, by the fundamental theorem of Calculus, one can easily see that M ϕ ′ ,ψ ′ ;µ = C ϕ,ψ , which is defined by Assuming 7 times continuous differentiability, the equality problem of these means was solved by Losonczi [15]. The equality problem of means in various classes of two-variable means has been solved. We refer here to Losonczi's works [14], [15], [16], [17], [18] where the equality of two-variable means is characterized. A key idea in these papers, under high order differentiablity assumptions, is to calculate and then compare the partial derivatives of the means at points of the form (x, x). A similar problem, the mixed equality problem of quasiarithmetic and Lagrangian means was solved by Páles [24].
The aim of this paper is to study the equality problem of generalized quasiarithmetic means, i.e., to characterize those pairs of functions (f, g) and (F, G) such that holds. Due to the complexity of the problem, we will not solve it in its natural generality. In our final main results we consider the cases when the measure µ is either of the form δ 0 +δ 1 2 or is the Lebesgue measure on [0, 1]. For these two cases, we shall need sixth-order differentiability properties of the unknown functions f, g and F, G.

Basic results
Given a Borel probability measure µ on the interval [0, 1], we define the first moment and the nth centralized moment of µ by and Clearly, µ 0 = 1 and µ 1 = 0. It is also obvious that µ 2n ≥ 0 and equality can hold if and only if µ is the Dirac measure δ µ 1 . In order to describe the regularity conditions related to the two unknown functions f, g generating the mean M f,g;µ , we introduce some notations. The class C 0 (I) consists of all those pairs (f, g) of continuous functions f, g : I → R such that g ∈ CP(I) and f /g ∈ CM(I). For n ∈ N, we say that the pair (f, g) is in the class C n (I) if f, g are n-times continuously differentiable functions such that g ∈ CP(I) and the function f ′ g − f g ′ does not vanish anywhere on I. Obviously, this latter condition implies that f /g is strictly monotone, i.e., f /g ∈ CM(I).
For (f, g) ∈ C 2 (I), we also introduce the notation where the (i, j)-order Wronskian operator W i,j is defined in terms of the ith and jth derivatives by Our first result establishes a formula for the higher-order derivatives of f and g as well as for their (i, j)-order Wronskian in terms of the functions Φ f,g and Ψ f,g .
where n ≥ 2 and define two sequences (ϕ i ) and (ψ i ) by the recursions In particular, Proof. Consider the second-order linear differential equation for the unknown function h : I → R. Obviously, (8) is satisfied for h ∈ {f, g}. We can see that (8) is equivalent to the equation i.e., h ∈ {f, g} is also a solution of (9). Observe that (9) is the i = 2 particular case of (5).) The equality (5) trivially holds if i = 0. Assume that (5) has been proved for some i ∈ {0, . . . , n − 1}. Then, using (5), (9), and (4), we get The equality in (6) follows from (5). Indeed, applying (5) for h = f and h = g, we obtain Hence, the proof of the lemma is complete.
In the sequel we shall need the first few members of the sequences (ϕ i ) and (ψ i ) constructed in (4). For the sake of convenience, we list, for small i, the first few members of them: We say that two pairs of functions (f, g), (F, G) ∈ C 0 (I) are equivalent, denoted by (f, g) ∼ (F, G), if there exists a nonsingular 2 × 2-matrix A (with real entries) such that In other words, (f, g) ∼ (F, G) holds if there exist four real constants a, b, c, d with ad = bc such that F = af + bg and G = cf + dg.
The remaining auxiliary results of this section were obtained in [19] and [25]. The property of equivalence in the class C 2 (I) is completely characterized by the following result.
The next result characterizes the mean M f,g;µ via an implicit equation.
As a consequence, the next lemma shows that the equivalent pairs of generating functions determine identical means.
3. Higher-order directional derivatives of generalized quasiarithmetic means Lemma 5. Let n ∈ N, (f, g) ∈ C n (I) and µ be a Borel probability measure on [0, 1]. Then M f,g;µ is n-times continuously differentiable on I × I.
Proof. The proof of this statement requires the use of standard calculus rules and a standard argument. One can verify that the inverse of f /g and the maps (x, y) are n-times differentiable on (f /g)(I) and I 2 , respectively.
In what follows, we deduce explicit formulae for the high-order directional derivatives of M f,g;µ at the diagonal points of the Cartesian product I × I. Given (f, g) ∈ C 0 (I) and a fixed element x ∈ I, define the function m x = m x;f,g;µ in a neigborhood of zero by whereμ 1 denotes the first moment of the measure µ.

Necessary and sufficient conditions for the equality of generalized quasiarithmetic means
In what follows, given (f, g), (F, G) ∈ C 0 (I) and a probability measure µ on [0, 1], we say that M f,g;µ equals M F,G;µ if they coincide at every point of I 2 . We say that these two means are equal x;F,G;µ (0).
Proof. Let U ⊆ I 2 be an open set containing a dense subset D of ∆(I) such that the two means are equal at every point of U . Let x ∈ I be fixed such that (x, x) ∈ D. Define Then U x is a neighbourhood of 0 (because U is open), and the equality of the means on U implies that, for any u ∈ U x , m x;f,g;µ (u) = m x;F,G;µ (u). Therefore (16) holds for all k ∈ {1, . . . , n} and x ∈ I with (x, x) ∈ D. Using the continuity, the density of D yields that this equality holds for all x ∈ I.
Proof. The condition µ 3 = 0 implies that µ 2 > 0 is also valid. Thus, the first equality in (20) is a consequence of Lemma 8. Applying the third-order formula of Proposition 6, the second equality (19) implies , which, using µ 3 = 0 and the equality Φ f,g = Φ F,G , proves the last equation in (20).
Proof. The condition µ 2 > 0 implies that µ 4 > 0 is also valid. It follows from Lemma 8 and the first condition in (21) that Φ f,g = Φ F,G holds on I. Using the formula for the fourth-order derivative by Proposition 6, the second equality in (21) simplifies to , Therefore, we get the following first-order homogeneous linear differential equation for the difference function R := Ψ f,g − Ψ F,G : The solution of this differential equation implies (22) for some γ ∈ R.
Proof. The condition µ 5 = 0 implies that µ 2 µ 4 = 0 is also valid. Therefore, by Lemma 8 and Lemma 11, the first two conditions in (25) yield that Φ f,g = Φ F,G and, with the notation R := Ψ f,g − Ψ F,G , the equalities in (22) and (24) hold for some γ ∈ R. Then From Lemma 11 it follows that R is either identically zero or nowhere zero in I. In the first case, we have that Ψ f,g = Ψ F,G . Thus, in the rest of the proof, we may assume that R is nowhere zero in I, i.e., γ = 0. Using the formula for the fifth-order derivative by Proposition 6, the third condition of (25) simplifies to Define S := Ψ f,g + Ψ F,G . Then the above equality yields Using (27), we finally get that which was to be proved.
If µ 6 = 5µ 2 µ 4 , then a simple computation shows that p, therefore we have that p + q + 1 = 0 and the coefficient of S ′ in the equality (34) is also not zero. From the definitions of p and q it follows that µ 4 = 3µ 2 2 p + 1 and µ 6 = 15µ 3 2 (q − p + 1) (p + 1)(p + q + 1) , respectively. Substituting these expressions into (34) and then multiplying it by − (p + 1)(p + q + 1) 90µ 3 2 p , we arrive at This is an inhomogeneous first-order linear differential equation for S, whose general solution is of the following form This equality combined with (22) and (36) completes the proof of the last assertion of the lemma.
As the main applications the above corollary, we restate and reprove the solution of the equality problems related to Bajraktarević and Cauchy means in the following two subsections.
For a real parameter t ∈ R, introduce the sine and cosine type functions S t , C t : R → R by It is easily seen that the functions S t and C t form the fundamental system of solutions for the secondorder homogeneous linear differential equation h ′′ = th.

4.1.
Equality of Bajraktarević means. The first main result of this section is a rephrased form of the result of Losonczi [14,18] which characterized the equality of Bajraktarević means. In these papers Losonczi established 1+32 cases for the equality of these means. To deduce the result of Losonczi from the theorem below, the best is to elaborate condition (vi) where, beyond the canonical case (that is the equivalence of the generating functions), the equality is described in terms of two polynomials of at most second degree. In the subcases when, independently, these polynomials are constants, of first degree, of second degree with no, or with one or with two real roots, we can distinguish 6 × 6 = 36 subcases which then reduce to the cases considered by Losonczi. (iii) For all x ∈ I, the equalities in (28) are satisfied.
(iv) Φ f,g = Φ F,G holds on I and there exist real constants α and β such that hold on I. (v) Either (f, g) ∼ (F, G) or there exist real constants a, b, c, A, B, C, γ such that and and W 1,0 F,G = γW 1,0 f,g . (vi) Either (f, g) ∼ (F, G) or there exist two real polynomials P and Q of at most second degree which are positive on the range of f /g and F/G, respectively, and there exist real constants γ and δ such that (vii) Either (f, g) ∼ (F, G) or there exist a strictly monotone function ϕ : I → R and real constants α and β such that Proof. The implication (i)⇒(ii) is obvious. The implication (ii)⇒(iii) is a direct consequence of Lemma 7.
To prove the implication (iv)⇒(v), assume that (iv) holds for some constants α, β ∈ R. If α = β, then Lemma 2 implies that (f, g) ∼ (F, G). Now consider the case when α = β. The existence of some γ such that the identity W 1,0 F,G = γW 1,0 f,g be valid is a direct consequence of the integration of the equality Φ f,g = Φ F,G . Applying implication (iv)⇒(ii) of [26,Theorem 10], we conclude that there exist real constants a, b, c, A, B, C such that the equalities (39) hold. Therefore, assertion (v) is valid.
Assume now that (v) holds for some constants a, b, c, A, B, C, γ and define Then, dividing the equalities in (39) side by side by g 2 and by G 2 , we obtain that Therefore, P and Q are positive on the codomain of f /g and F/G, respectively, and the first two equalities in (40) hold. Furthermore, using (41), we have that and similarly, Applying the equality W 1,0 F,G = γW 1,0 f,g , after integration we obtain that the third equality in (40) is also valid for some real constant δ. This shows that assertion (v) implies (vi).
Reversing the steps of the previous argument, one can easily see that assertion (vi) also implies (v), where a, b, c and A, B, C are the coefficients of the polynomials P and Q, respectively.

4.2.
Equality of Cauchy means. The second main result of this section is a rephrased form of the results of Losonczi [15,17] which characterized the equality of Cauchy means and established 1+32 cases for the equality of these means. The results of Losonczi can easily be deduced from condition (vi) of the next theorem where, beyond the canonical case the equality is described in terms of two polynomials of at most second degree. Considering the same subcases as for Theorem 15, one can again distinguish 6 × 6 = 36 subcases which then reduce to the cases considered by Losonczi. (iii) For all x ∈ I, the equalities in (28) are satisfied.
(iv) Φ f,g = Φ F,G and there exist constants α, β ∈ R such that hold on I. (v) Either (f, g) ∼ (F, G) or there exist real constants a, b, c, A, B, C, γ such that af 2 + bf g + cg 2 = W 1,0 f,g 2 3 and and W 1,0 F,G = γW 1,0 f,g . (vi) Either (f, g) ∼ (F, G) or there exist two real polynomials P and Q of at most second degree which are positive on the range of f /g and F/G, respectively, and there exist real constants γ and δ such that (vii) Either (f, g) ∼ (F, G) or there exist a strictly monotone differentiable function ϕ : I → R and real constants α and β such that Proof. The implication (i)⇒(ii) is clear. The implication (ii)⇒(iii) is a direct consequence of Lemma 7. Assume now that assertion (iii) is valid. Using that µ is the Lebesgue measure restricted to [0, 1], it is easily seen that Consequently, conditions µ 2 > 0 = µ 3 , µ 4 = 3µ 2 2 and 6µ 6 µ 2 2 − µ 6 µ 4 − 5µ 2 4 µ 2 = 0 are valid and using (23), we get p = 2 3 . Hence the second alternative of Corollary 14 is applicable. Thus Φ f,g = Φ F,G holds and the equalities in (38) reduce to (42), which completes the proof of assertion (iv).
To prove the implication (iv)⇒(v), assume that (iv) holds for some constants α, β ∈ R. If α = β, then Lemma 2 implies that (f, g) ∼ (F, G). Now consider the case when α = β. The existence of some γ such that the identity W 1,0 F,G = γW 1,0 f,g be valid is a direct consequence of the integration of the equality Φ f,g = Φ F,G . The equalities in (42) are constants. Therefore, applying implication (vii)⇒(iv) of [21,Theorem 7], we conclude that there exist real constants a, b, c, A, B, C that validate assertion our (v). Assume now that (v) holds for some constants a, b, c, A, B, C, γ and define P (t) := at 2 + bt + c and Q(t) := At 2 + Bt + C (t ∈ R).
Then, dividing the equalities in (39) side by side by g 2 and by G 2 , we obtain that Therefore, P and Q are positive on the codomain of f /g and F/G, respectively. Using the identities W 1,0 f,g = g 2 (f /g) ′ and W 1,0 F,G = G 2 (F/G) ′ , the above equalities yield the first two equations in (43). Furthermore, using (45), we have that Applying the equality W 1,0 F,G = γW 1,0 f,g , after integration we obtain that the third equality in (43) is also valid for some real constant δ. This completes the proof of the implication (v)⇒(vi).
Reversing the steps of the previous argument, one can easily see that assertion (vi) also implies (v), where a, b, c and A, B, C are the coefficients of the polynomials P and Q, respectively.
The implications (viii)⇒(ix) and (ix)⇒(i) are obvious. Finally, the equivalence of the assertions (vii) and (ix) is a consequence of the main result of the paper [12].

4.3.
Conclusion and open problems. We have to stress that the different assertions of Theorem 15 and Theorem 16 require different order of regularity. Obviously, assertions (i), (ii), (vi), (vii), (ix) make sense in the regularity class C 0 (I). For (v) and (viii) one has to take the unknown functions from C 1 (I). Finally, assertions (iv) and (iii) require the regularity class C 2 (I) and C 6 (I), respectively.
One can also see that some of the implications described in the above proof are valid with smaller order regularity assumptions. For instance, (i) implies (ii) and (ix) implies (i) in the class C 0 (I). For the implications (ii)⇒(iii)⇒(iv), we need C 6 (I). The proof of the implication (iv)⇒(v) requires C 2 (I), while the equivalence of assertions (v) and (vi), and the implications (v)⇒(viii)⇒(ix) can be verified in the regularity class C 1 (I).
Based on the above observations, we can formulate the following three open problems.