Abstract
Given two functions \(f,g:I\rightarrow \mathbb {R}\) and a probability measure \(\mu \) on the Borel subsets of [0, 1], the twovariable mean \(M_{f,g;\mu }:I^2\rightarrow I\) is defined by
This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević 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) for which
holds. Under at most sixthorder differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarević means and of Cauchy means.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 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 realvalued functions defined on I will be denoted by \(\mathscr {CM}(I)\) and \(\mathscr {CP}(I)\), respectively.
In general, a continuous function \(M:I^2\rightarrow I\) is called a twovariable mean on I if the socalled mean value inequality
holds. If, for \(x\ne y\), both of the inequalities in (1) are strict, then M is called a twovariable strict mean. Arithmetic and geometric means are well known instances of strict means on \(\mathbb {R}_+\).
Given a function \(\varphi \in \mathscr {CM}(I)\), the twovariable quasiarithmetic mean generated by \(\varphi \) is the function \(A_\varphi :I^2\rightarrow I\) defined by
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 \(A_\varphi \) and \(A_\psi \) are equal to each other if and only if there exist two real constants \(a\ne 0\) and b such that \(\psi =a\varphi +b\).
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 nonfixed. For the twovariable case, Aczél [1,2,3,4] (see also [5]), proved a characterization theorem involving the notion of bisymmetry. This result was extended to the nvariable 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 \(f,g:I\rightarrow \mathbb {R}\) with \(g\in \mathscr {CP}(I)\), \(f/g\in \mathscr {CM}(I)\) and a probability measure \(\mu \) on the Borel subsets of [0, 1], the twovariable mean \(M_{f,g;\mu }:I^2\rightarrow I\) is defined by
Means of the above form, will be called generalized quasiarithmetic means.
The first important particular case of this definition is when \(\mu =\frac{\delta _0+\delta _1}{2}\). Here and in the sequel, \(\delta _\tau \) will denote the Dirac measure concentrated at the point \(\tau \in [0,1]\). If \(\varphi \in \mathscr {CM}(I)\), and \(p\in \mathscr {CP}(I)\), then \(M_{\varphi \cdot p,p;\mu }=B_{\varphi ,p}\), which is defined by
and is called the twovariable 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 \(\mu \) is the Lebesgue measure on [0, 1] and \(\varphi ,\psi :I\rightarrow I\) are continuously differentiable functions with \(\psi '\in \mathscr {CP}(I)\) and \(\varphi '/\psi '\in \mathscr {CM}(I)\). Then, by the fundamental theorem of Calculus, one can easily see that \(M_{\varphi ',\psi ';\mu }=C_{\varphi ,\psi }\), which is defined by
Assuming 7 times continuous differentiability, the equality problem of these means was solved by Losonczi [17].
The equality problem of means in various classes of twovariable means has been solved. We refer here to Losonczi’s works [14,15,16,17,18] where the equality of twovariable means is characterized. The key idea in these papers, under high order differentiability assumptions, is to calculate and then compare the partial derivatives of the means at points of the diagonal. A similar problem, the mixed equality problem of quasiarithmetic and Lagrangian means was solved by Páles [24]. Another mixed equality problem, the equality of twovariable quasiarithmetic and Bajraktarevic means in the symmetric and in the weighted setting was also solved by Daróczy–Maksa–Páles [8] and by Kiss–Páles [11], respectively.
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) for which
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 \(\mu \) is either of the form \(\frac{\delta _0+\delta _1}{2}\) or is the Lebesgue measure on [0, 1]. For these two cases, we shall need sixthorder differentiability properties of the unknown functions f, g and F, G.
2 Basic results
Given a Borel probability measure \(\mu \) on the interval [0, 1], we define the first moment and the nth centralized moment of \(\mu \) by
Clearly, \(\mu _0=1\) and \(\mu _1=0\). It is also obvious that \(\mu _{2n}\ge 0\) and equality can hold if and only if \(\mu \) is the Dirac measure \(\delta _{\widehat{\mu }_1}\).
In order to describe the regularity conditions related to the two unknown functions f, g generating the mean \(M_{f,g;\mu }\), we introduce some notations. The class \(\mathscr {C}_0(I)\) consists of all pairs (f, g) of continuous functions \(f,g:I\rightarrow \mathbb {R}\) such that \(g\in \mathscr {CP}(I)\) and \(f/g\in \mathscr {CM}(I)\). For \(n\in \mathbb {N}\), we say that the pair (f, g) is in the class \(\mathscr {C}_n(I)\) if f, g are ntimes continuously differentiable functions such that \(g\in \mathscr {CP}(I)\) and the function \(f'gfg'\) does not vanish anywhere on I. Obviously, this latter condition implies that f/g is strictly monotone, i.e., \(f/g\in \mathscr {CM}(I)\).
In what follows, the symbol \(f^{(i)}\) will stand for the ith derivative of the function \(f:I\rightarrow \mathbb {R}\), where \(i\in \mathbb {N}\cup \{0\}\). In the particular cases \(i\in \{0,1,2\}\), we may use the alternative notations \(f,f',f''\), respectively. For \((f,g)\in \mathscr {C}_2(I)\), 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 higherorder derivatives of f and g as well as for their (i, j)order Wronskian in terms of the functions \(\Phi _{f,g}\) and \(\Psi _{f,g}\).
Lemma 1
Let \((f,g)\in \mathscr {C}_n(I)\), where \(n\ge 2\) and define two sequences \((\varphi _i)\) and \((\psi _i)\) by the recursions
Then, for \(h\in \{f,g\}\), that is, for \(h=f\) and \(h=g\),
and
In particular,
Proof
Consider the secondorder linear differential equation
for the unknown function \(h:I\rightarrow \mathbb {R}\). Obviously, (8) is satisfied for \(h\in \{f,g\}\). We can see that (8) is equivalent to the equation
i.e., \(h\in \{f,g\}\) is also a solution of (9). (Observe that (9) is a particular case of (5) when \(i=2\).)
The equality (5) trivially holds if \(i=0\). Assume that (5) has been proved for some \(i\in \{0,\dots ,n1\}\). Then, using (5), (9), and (4), we get
which completes the proof of the induction.
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. \(\square \)
In the sequel we shall need the first few members of the sequences \((\varphi _i)\) and \((\psi _i)\) constructed in (4). For the sake of convenience, we list their first few members for small i:
and
We say that two pairs of functions \((f,g),(F,G)\in \mathscr {C}_0(I)\) are equivalent, denoted \((f,g)\sim (F,G)\), if there exists a nonsingular \(2\times 2\)matrix A (with real entries) such that
In other words, \((f,g)\sim (F,G)\) holds if there exist four real constants a, b, c, d with \(ad\ne bc\) 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 \(\mathscr {C}_2(I)\) is completely characterized by the following result.
Lemma 2
[25, Theorem 2.1] Let \((f,g),(F,G)\in \mathscr {C}_2(I)\). Then \((f,g)\sim (F,G)\) holds if and only if
The next result characterizes the mean \(M_{f,g;\mu }\) via an implicit equation.
Lemma 3
([19, Lemma 1], [25, Lemma 1.1]) Let \((f,g)\in \mathscr {C}_0(I)\) and \(\mu \) be a Borel probability measure on [0, 1]. Then for all \(x,y\in I\), the value \(z=M_{f,g;\mu }(x,y)\) is the unique solution of the equation
As a consequence, the next lemma shows that equivalent pairs of generating functions determine identical means.
Lemma 4
[19, 25] Let \((f,g),(F,G)\in \mathscr {C}_0(I)\) and \(\mu \) be a Borel probability measure on [0, 1]. Assume that \((f,g)\sim (F,G)\). Then \(M_{F,G;\mu }=M_{f,g;\mu }\).
3 Higherorder directional derivatives of generalized quasiarithmetic means
Lemma 5
Let \(n\in \mathbb {N}\), \((f,g)\in \mathscr {C}_n(I)\) and \(\mu \) be a Borel probability measure on [0, 1]. Then \(M_{f,g;\mu }\) is ntimes continuously differentiable on \(I\times I\).
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 \((x,y)\mapsto \int _0^1 h(tx+(1t)y)d\mu (t)\) (where \(h\in \{f,g\}\)) are ntimes differentiable on (f/g)(I) and \(I^2\), respectively. \(\square \)
In what follows, we deduce explicit formulae for the highorder directional derivatives of \(M_{f,g;\mu }\) at the diagonal points of the Cartesian product \(I\times I\). Given \((f,g)\in \mathscr {C}_0(I)\) and a fixed element \(x\in I\), define the function \(m_x=m_{x;f,g;\mu }\) in a neigborhood of zero by
where \(\hat{\mu }_1\) denotes the first moment of the measure \(\mu \).
Lemma 6
Let \(n\in \mathbb {N}\), \((f,g)\in \mathscr {C}_n(I)\), and \(\mu \) be a Borel probability measure on [0, 1]. Then, for fixed \(x\in I\), the function \(m_x\) defined by (13) is ntimes continuously differentiable at the origin, \(m_x(0)=x\) and in the cases \(n=1,2,3,4,5,6\), we have
and
Proof
Let \(x\in I\) be fixed. Then the ntimes continuous differentiability of \(m_x\) at the origin is a consequence of Lemma 5. By Lemma 3, for small u, we have the equality
Differentiating this equality ntimes with respect to u and using Leibniz’s rule, we obtain
Now the substitution \(u=0\) implies
The equality \(m_x(0)=x\) follows from the definition of \(m_x\). Since \(\mu _0=1\), \(\mu _1=0\), in the case \(n=1\), (14) reduces to
which yields \(m_x'(0)=0\).
To elaborate the condition (14) in the cases \(n=2,3,4,5,6\), we shall need the following computational rules for the functions \(h\in \{f,g\}\),
Hence, using \(m_x(0)=x\) and \(m_x'(0)=0\), it follows that
In the case \(n=2\), using \(\mu _0=1\) and \(\mu _1=0\), Eq. (14) yields
which, in view of the identities (15), reduces to the equality
proving \(m_x^{(2)}(0)=\mu _2\Phi _{f,g}(x)\).
In the case \(n=3\), using \(\mu _0=1\), \(\mu _1=0\), and \(m'(0)=0\), Eq. (14) gives
which, applying the third identity in (15) for \(h=f\) and \(h=g\), implies
Hence, using (7) for \(i=3\), we get
In the case \(n=4\), using \(\mu _0=1\), \(\mu _1=0\) and \(m'(0)=0\), Eq. (14) results in
which, by the second and fourth identities in (15), implies
Thus, applying (7) and the formulae for \(m_x^{(2)}(0)\), \(\varphi _2,\varphi _4\), and \(\psi _2\), we obtain
In the case \(n=5\), using \(\mu _0=1\), \(\mu _1=0\) and \(m'(0)=0\), Eq. (14) results in
which, by the identities (15), implies
Thus, applying (7), the above equality yields that
Using the explicit formulae for \(m_x^{(2)}(0)\), \(m_x^{(3)}(0)\), \(\varphi _2,\varphi _5\), and \(\psi _2,\psi _3\), this expression simplifies to the required statement.
Finally, in the case \(n=6\), using \(\mu _0=1\), \(\mu _1=0\) and \(m'(0)=0\), Eq. (14) results in
which, using the identities (15), implies
Thus, applying (7), we get
Using the explicit formulae for \(m_x^{(2)}(0),m_x^{(3)}(0),m_x^{(4)}(0)\), \(\varphi _2,\varphi _3,\varphi _6\), and for \(\psi _2,\psi _3,\psi _4\), this formula simplifies to the desired statement. \(\square \)
4 Necessary and sufficient conditions for the equality of generalized quasiarithmetic means
In what follows, given \((f,g),(F,G)\in \mathscr {C}_0(I)\) and a probability measure \(\mu \) on [0, 1], we say that \(M_{f,g;\mu }\) equals \(M_{F,G;\mu }\) if they coincide at every point of \(I^2\). We say that these two means are equal near the diagonal \(\Delta (I):=\{(x,x)\mid x\in I\}\) of \(I^2\), if there exists an open set \(U\subseteq I^2\) containing a dense subset D of \(\Delta (I)\) such that the two means are equal at every point of U.
Lemma 7
Let \(\mu \) be a Borel probability measure on [0, 1], let \(n\in \mathbb {N}\) and let \((f,g),(F,G)\in \mathscr {C}_n(I)\). If \(M_{f,g;\mu }\) equals \(M_{F,G;\mu }\) near the diagonal of \(I^2\), then, for all \(k\in \{1,\dots ,n\}\) and \(x\in I\),
Proof
Let \(U\subseteq I^2\) be an open set containing a dense subset D of \(\Delta (I)\) such that the two means are equal at every point of U. Let \(x\in I\) be fixed so that \((x,x)\in D\). 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 \(u\in U_x\),
Therefore (16) holds for all \(k\in \{1,\dots ,n\}\) and \(x\in I\) with \((x,x)\in D\). Using the continuity, the density of D yields that this equality holds for all \(x\in I\). \(\square \)
In the subsequent lemmas, we will analyse the consequences of the equalities (16) for \(k\in \{2,3,4,5,6\}\).
Lemma 8
Let \(\mu \) be a Borel probability measure on [0, 1] with \(\mu _2>0\) and let \((f,g),(F,G)\in \mathscr {C}_2(I)\). If, for all \(x\in I\),
holds, then
Proof
Applying the secondorder formula of Lemma 6, the equality (17) implies
which, using \(\mu _2>0\), proves (18). \(\square \)
Lemma 9
Let \(\mu \) be a Borel probability measure on [0, 1] with \(\mu _3\ne 0\) and let \((f,g),(F,G)\in \mathscr {C}_3(I)\). If, for all \(x\in I\),
hold, then
Proof
The condition \(\mu _3\ne 0\) implies that \(\mu _2>0\) is also valid. Thus, the first equality in (20) is a consequence of Lemma 8. Applying the thirdorder formula of Lemma 6, the second equality in (19) implies
which, using \(\mu _3\ne 0\) and the equality \(\Phi _{f,g}=\Phi _{F,G}\), proves the last equation in (20). \(\square \)
The following theorem characterizes the equality of generalized quasiarithmetic means under 3 times differentiability assumptions in the case when the third central moment is different from zero.
Theorem 10
Let \(\mu \) be a Borel probability measure on [0, 1] with \(\mu _3\ne 0\) and let \((f,g),(F,G)\in \mathscr {C}_3(I)\). Then the following assertions are equivalent:

(i)
The means \(M_{f,g;\mu }\) and \(M_{F,G;\mu }\) are equal on \(I^2\).

(ii)
The means \(M_{f,g;\mu }\) and \(M_{F,G;\mu }\) are equal near the diagonal of \(I^2\).

(iii)
For all \(x\in I\), the equalities in (19) hold.

(iv)
The equalities \(\Phi _{f,g}=\Phi _{F,G}\) and \(\Psi _{f,g}=\Psi _{F,G}\) hold on I.

(v)
\((f,g)\sim (F,G)\) holds.
Proof
The implication (i) \(\Rightarrow \) (ii) is obvious. The implication (ii) \(\Rightarrow \) (iii) is a consequence of Lemma 7. The implication (iii) \(\Rightarrow \) (iv) follows from Lemma 9. Finally, the implications (iv) \(\Rightarrow \) (v) and (v) \(\Rightarrow \) (i) are consequences of Lemma 2 and Lemma 4, respectively. \(\square \)
Lemma 11
Let \(\mu \) be a Borel probability measure on [0, 1] with \(\mu _2>0\) and let \((f,g),(F,G)\in \mathscr {C}_4(I)\). If, for all \(x\in I\),
hold, then the equality \(\Phi _{f,g}=\Phi _{F,G}=:\Phi \) holds on I and there exists a constant \(\gamma \in \mathbb {R}\) such that
where
Proof
The condition \(\mu _2>0\) implies that \(\mu _4>0\) is also valid. It follows from Lemma 8 and the first condition in (21) that \(\Phi _{f,g}=\Phi _{F,G}\) holds on I. Using the formula for the fourthorder derivative by Lemma 6, the second equality in (21) simplifies to
Therefore, we get the following firstorder homogeneous linear differential equation for the difference function \(R:=\Psi _{f,g}\Psi _{F,G}\):
The solution of this differential equation implies (22) for some \(\gamma \in \mathbb {R}\). \(\square \)
Lemma 12
Let \(\mu \) be a Borel probability measure on [0, 1] with \(\mu _3=0\ne \mu _5\) and let \((f,g),(F,G)\in \mathscr {C}_5(I)\). If, for all \(x\in I\),
hold, then the equality \(\Phi _{f,g}=\Phi _{F,G}=:\Phi \) holds on I and either \(\Psi _{f,g}=\Psi _{F,G}\) holds on I or there exists a nonzero real constant \(\gamma \) such that
where p is defined by (23).
Proof
The condition \(\mu _5\ne 0\) implies that \(\mu _2\mu _4\ne 0\) is also valid. Therefore, by Lemma 8 and Lemma 11, the first two conditions in (25) yield that \(\Phi _{f,g}=\Phi _{F,G}\) and, with the notation \(R:=\Psi _{f,g}\Psi _{F,G}\), the equalities in (22) and (24) hold for some \(\gamma \in \mathbb {R}\). Then
From Lemma 11 it follows that R is either identically zero or nowhere zero in I. In the first case, we have that \(\Psi _{f,g}=\Psi _{F,G}\). Thus, in the rest of the proof, we may assume that R is nowhere zero in I, i.e., \(\gamma \ne 0\).
Using the formula for the fifthorder derivative by Lemma 6, the third condition of (25) simplifies to
Define \(S:=\Psi _{f,g}+\Psi _{F,G}\). Then the above equality yields
Using (27), we finally get that
Therefore,
Thus
which was to be proved. \(\square \)
Lemma 13
Let \(\mu \) be a Borel probability measure on [0, 1] with \(\mu _2>0=\mu _3\) and let \((f,g),(F,G)\in \mathscr {C}_6(I)\). If, for all \(x\in I\),
hold, then \(\Phi _{f,g}=\Phi _{F,g}=:\Phi \) and either \(\Psi _{f,g}=\Psi _{F,g}\) or there exists a nonzero real constant \(\gamma \) such that (22) holds with p defined by (23) and we have the following alternatives:

(1)
If \(\mu _6=5\mu _2\mu _4\) and \(\mu _4=3\mu _2^2\), then \(\Phi \) is an at most first degree polynomial and \(\Psi _{f,g}\Psi _{F,g}\) is constant on I.

(2)
If \(\mu _6=5\mu _2\mu _4\) and \(\mu _4\ne 3\mu _2^2\), then the equalites
$$\begin{aligned} \Psi _{f,g}&=\gamma \big W_{f,g}^{1,0}\big ^p \tfrac{p+1}{3p}\tfrac{\Phi ^{(2)}}{\Phi }\tfrac{2p+3}{2}\Phi ' \tfrac{2p^2+3p+4}{6}\Phi ^2, \\ \Psi _{F,G}&=\gamma \big W_{f,g}^{1,0}\big ^p \tfrac{p+1}{3p}\tfrac{\Phi ^{(2)}}{\Phi }\tfrac{2p+3}{2}\Phi ' \tfrac{2p^2+3p+4}{6}\Phi ^2 \end{aligned}$$hold on the subset \(I\setminus \Phi ^{1}(0)=\{x\in I\mid \Phi (x)\ne 0\}\).

(3)
If \(\mu _6\ne 5\mu _2\mu _4\) and \(\mu _4=3\mu _2^2\), then there exists a real constant \(\delta \) such that the equalities
$$\begin{aligned} \Psi _{f,g}&=\gamma +\delta \big W_{f,g}^{1,0}\big ^{1}\!\tfrac{r}{2}\Phi ' +\tfrac{r5}{4} \Phi ^2\tfrac{3r7}{12}\big W_{f,g}^{1,0}\big ^{1} \bigg (\!\int \!\! \Phi ^3\big W_{f,g}^{1,0}\big \!\bigg ), \\ \Psi _{F,G}&=\gamma +\delta \big W_{f,g}^{1,0}\big ^{1}\!\tfrac{r}{2}\Phi ' +\tfrac{r5}{4} \Phi ^2\tfrac{3r7}{12}\big W_{f,g}^{1,0}\big ^{1} \bigg (\!\int \!\! \Phi ^3\big W_{f,g}^{1,0}\big \!\bigg ) \end{aligned}$$hold on I, where
$$\begin{aligned} r:=\dfrac{7\mu _645\mu _2^3}{3\mu _645\mu _2^3}. \end{aligned}$$(29) 
(4)
If \(\mu _6\ne 5\mu _2\mu _4\) and \(\mu _4\ne 3\mu _2^2\), then there exists a real constant \(\delta \) such that the equalities
$$\begin{aligned} \Psi _{f,g}&=\gamma \big W_{f,g}^{1,0}\big ^p +\delta \big W_{f,g}^{1,0}\big ^q+\tfrac{2(qp)(p+1)p+2}{6p}\Phi '\\&\quad +\tfrac{(qp)(q+3p+1)(p+1)+p^22p}{6p} \Phi ^2\\&\quad +\tfrac{(qp)(2p+q)(p+q+1)(p+1)}{6p}\cdot \big W_{f,g}^{1,0}\big ^q \bigg (\int \Phi ^3\big W_{f,g}^{1,0}\big ^{q}\bigg ), \\ \Psi _{F,G}&=\gamma \big W_{f,g}^{1,0}\big ^p +\delta \big W_{f,g}^{1,0}\big ^q+\tfrac{2(qp)(p+1)p+2}{6p}\Phi '\\&\quad +\tfrac{(qp)(q+3p+1)(p+1)+p^22p}{6p} \Phi ^2\\&\quad +\tfrac{(qp)(2p+q)(p+q+1)(p+1)}{6p}\cdot \big W_{f,g}^{1,0}\big ^q \bigg (\int \Phi ^3\big W_{f,g}^{1,0}\big ^{q}\bigg ) \end{aligned}$$hold on I, where
$$\begin{aligned} q:=\frac{\mu _2}{\mu _4}\cdot \frac{10\mu _4^23\mu _6\mu _215\mu _4\mu _2^2}{\mu _65\mu _4\mu _2}. \end{aligned}$$(30)
Proof
The condition \(\mu _2>0\) implies that \(\mu _4>0\) and \(\mu _6>0\) are also valid. The equalities \(\Phi _{f,g}=\Phi _{F,g}\) and (22) are consequences of Lemma 8 and Lemma 11, respectively. If \(\gamma =0\), then \(\Psi _{f,g}=\Psi _{F,g}\). Therefore, in the rest of the proof, we may assume that \(\gamma \) is not zero.
Using the formula for the sixthorder derivative by Lemma 6, the third condition of (28), we arrive at
Thus, introducing \(R:=\Psi _{f,g}\Psi _{F,G}\) and using the function \(S:=\Psi _{f,g}+\Psi _{F,G}\) defined earlier, we can rewrite this equation as
On the other hand, using (24), (27), and \(R^{(3)}=(p\Phi ^{(2)}+3p^2\Phi '\Phi +p^3\Phi ^3)R\), we obtain
Since \(\gamma \ne 0\), the function R is nowhere zero and we get the following equation for S and \(\Phi \):
Consider first the case when \(\mu _4=3\mu _2^2\). Then \(p=0\) and (32) simplifies to
If additionally \(\mu _6=5\mu _2\mu _4=15\mu _2^3\), then
which yields that \(\Phi \) is an at most first degree polynomial. This, together with the result of Lemma 11, completes the proof of assertion (i). On the other hand, if \(\mu _6\ne 5\mu _2\mu _4=15\mu _2^3\), then dividing the Eq. (33) by \(3(\mu _615\mu _2^3)\ne 0\), we get
where r is defined in (29). This is a firstorder inhomogeneous linear differential equation for S, whose general solution is of the form
where \(\delta \) is an arbitrary real constant. Using the equalities
and (22), assertion (iii) follows directly.
Now consider the case when \(\mu _4\ne 3\mu _2^2\) which is equivalent to \(p\ne 0\). If \(\mu _6=5\mu _2\mu _4\), then the coefficient of \(S'\) in the equality (32) is zero. Using the definition (23) of p and the equality \(\mu _6=5\mu _2\mu _4\), (32) can be rewritten as
This implies that
Therefore, using (34) and (22), the second assertion follows.
If \(\mu _6\ne 5\mu _2\mu _4\), then a simple computation shows that
therefore we have that \(p+q+1\ne 0\) and the coefficient of \(S'\) in the equality (32) is not zero either. From the definitions of p and q it follows that
respectively. Substituting these expressions into (32) and then multiplying it by \(\tfrac{(p+1)(p+q+1)}{90\mu _2^3p}\), we arrive at
This is an inhomogeneous firstorder linear differential equation for S, whose general solution is of the following form
This equality combined with (22) and (34) completes the proof of the last assertion of the lemma. \(\square \)
Corollary 14
Let \(\mu \) be a Borel probability measure on [0, 1] with \(\mu _2>0=\mu _3\) and \(6\mu _6\mu _2^2\mu _6\mu _45\mu _4^2\mu _2=0\) and let \((f,g),(F,G)\in \mathscr {C}_6(I)\). If, for all \(x\in I\), the equalities in (28) are satisfied, then \(\Phi _{f,g}=\Phi _{F,g}=:\Phi \) and we have the following alternatives

(1)
If \(\mu _4=3\mu _2^2\), then \(p=0\), \(\Phi \) is an at most first degree polynomial and \(\Psi _{f,g}\Psi _{F,G}\) is a constant.

(2)
If \(\mu _4\ne 3\mu _2^2\), then \(p\ne 0\) and there exist real constants \(\alpha \) and \(\beta \) such that the equalities
$$\begin{aligned} \Psi _{f,g}&=\alpha \big W_{f,g}^{1,0}\big ^p +\tfrac{2p}{6p}\Phi '+\tfrac{p2}{6}\Phi ^2 \quad \text{ and }\nonumber \\ \Psi _{F,G}&=\beta \big W_{f,g}^{1,0}\big ^p +\tfrac{2p}{6p}\Phi '+\tfrac{p2}{6} \Phi ^2 \end{aligned}$$(35)hold on I, where p is given by (23).
Proof
The equality \(\Phi _{f,g}=\Phi _{F,g}\) is a consequence of Lemma 8.
If \(\mu _4=3\mu _2^2\), then the moment condition \(6\mu _6\mu _2^2\mu _6\mu _45\mu _4^2\mu _2=0\) implies that \(\mu _6=5\mu _4\mu _2\). Hence, we are in the alternative (i) of Lemma 13, which yields that \(\Phi \) is an at most first degree polynomial and \(\Psi _{f,g}\Psi _{F,G}\) is a constant on I.
If \(\mu _4\ne 3\mu _2^2\), then \(p\ne 0\) and the moment condition \(6\mu _6\mu _2^2\mu _6\mu _45\mu _4^2\mu _2=0\) implies that \(\mu _6\ne 5\mu _4\mu _2\) and \(p=q\), where q is defined by (30). Now we satisfy the conditions of the alternative (iv) of Lemma 13, hence (35) is valid with \(\alpha :=\gamma +\delta \) and \(\beta :=\delta \gamma \). \(\square \)
As the main applications of 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 \(t\in \mathbb {R}\), introduce the sine and cosine type functions \(S_t,C_t:\mathbb {R}\rightarrow \mathbb {R}\) 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 \(h^{(2)}=th\).
4.1 Equality of Bajraktarević means
The first main result of this section is a rephrased form of results of Losonczi [14, 18] who characterized the equality of Bajraktarević means. In these papers Losonczi established the canonical case and additionally 32 cases for the equality of these means. To deduce the result of Losonczi from the theorem below, the most efficient way is to elaborate condition (vi). There, 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 real roots, or with one or two real roots, we can distinguish \(6\times 6=36\) subcases which then reduce to the 32 cases considered by Losonczi.
Theorem 15
Let \((f,g),(F,G)\in \mathscr {C}_6(I)\) and let \(\mu :=\frac{\delta _0+\delta _1}{2}\). Then the following assertions are equivalent:

(i)
The means \(M_{f,g;\mu }\) and \(M_{F,G;\mu }\) are equal on \(I^2\).

(ii)
The means \(M_{f,g;\mu }\) and \(M_{F,G;\mu }\) are equal near the diagonal of \(I^2\).

(iii)
For all \(x\in I\), the equalities in (28) are satisfied.

(iv)
\(\Phi _{f,g}=\Phi _{F,G}\) holds on I and there exist real constants \(\alpha \) and \(\beta \) such that
$$\begin{aligned} \Psi _{f,g}=\alpha \big (W_{f,g}^{1,0}\big )^2 \quad \text{ and }\quad \Psi _{F,G}=\beta \big (W_{f,g}^{1,0}\big )^2 \end{aligned}$$hold on I.

(v)
Either \((f,g)\sim (F,G)\) or there exist real constants \(a,b,c,A,B,C,\gamma \) such that
$$\begin{aligned} af^2+bfg+cg^2=1 \quad \text{ and }\quad AF^2+BFG+CG^2=1 \end{aligned}$$(36)and \(W_{F,G}^{1,0}=\gamma W_{f,g}^{1,0}\).

(vi)
Either \((f,g)\sim (F,G)\) 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 \(\gamma \) and \(\delta \) such that
$$\begin{aligned}&g=\frac{1}{\sqrt{P}}\!\circ \!\frac{f}{g}, \quad G=\frac{1}{\sqrt{Q}}\!\circ \!\frac{F}{G},\quad \text{ and } \nonumber \\&\quad \quad \bigg (\!\int \frac{1}{Q}\!\bigg )\!\circ \!\frac{F}{G}=\gamma \bigg (\!\int \frac{1}{P}\!\bigg )\!\circ \!\frac{f}{g}+\delta . \end{aligned}$$(37) 
(vii)
Either \((f,g)\sim (F,G)\) or there exist a strictly monotone function \(\varphi :I\rightarrow \mathbb {R}\) and real constants \(\alpha \) and \(\beta \) such that
$$\begin{aligned} (f,g)\sim (S_\alpha \circ \varphi ,C_\alpha \circ \varphi ) \quad \text{ and }\quad (F,G)\sim (S_\beta \circ \varphi ,C_\beta \circ \varphi ). \end{aligned}$$ 
(viii)
Either \((f,g)\sim (F,G)\) or \(M_{f,g;\mu }=A_\varphi =M_{F,G;\mu }\) holds on \(I^2\) with \(\varphi :=\int W_{f,g}^{1,0}\).

(ix)
Either \((f,g)\sim (F,G)\) or there exists a strictly monotone function \(\varphi :I\rightarrow \mathbb {R}\) such that \(M_{f,g;\mu }=A_\varphi =M_{F,G;\mu }\) holds on \(I^2\).
Proof
The implication (i) \(\Rightarrow \) (ii) is obvious. The implication (ii) \(\Rightarrow \) (iii) is a direct consequence of Lemma 7. Assume now that assertion (iii) is valid. Using that \(\mu \) is of the form \(\dfrac{\delta _0+\delta _1}{2}\), an easy computation shows that
Therefore, the conditions \(\mu _2>0=\mu _3\), \(\mu _4\ne 3\mu _2^2\) and \(6\mu _6\mu _2^2\mu _6\mu _45\mu _4^2\mu _2=0\) hold, whence using (23), we get \(p=2\). Now the second alternative of Corollary 14 is applicable and it implies assertion (iv).
To prove the implication (iv) \(\Rightarrow \) (v), assume that (iv) holds for some constants \(\alpha ,\beta \in \mathbb {R}\). If \(\alpha =\beta \), then Lemma 2 implies that \((f,g)\sim (F,G)\). Now consider the case when \(\alpha \ne \beta \). The existence of some \(\gamma \) such that the identity \(W_{F,G}^{1,0}=\gamma W_{f,g}^{1,0}\) holds is a direct consequence of the integration of the equality \(\Phi _{f,g}=\Phi _{F,G}\). Applying implication (iv) \(\Rightarrow \) (ii) of [26, Theorem 10], we conclude that there exist real constants a, b, c, A, B, C such that the equalities (36) hold. Therefore, assertion (v) is valid.
Assume now that (v) holds for some constants \(a,b,c,A,B,C,\gamma \) and define
Then, dividing the equalities in (36) 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 (37) hold. Furthermore, using (38), we have that
and similarly,
Applying the equality \(W_{F,G}^{1,0}=\gamma W_{f,g}^{1,0}\), after integration we obtain that the third equality in (37) is also valid for some real constant \(\delta \). 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.
To prove the implication (v) \(\Rightarrow \) (viii), assume that (v) holds for some constants \(a,b,c,A,B,C,\gamma \in \mathbb {R}\). The implication (ii) \(\Rightarrow \) (iii) of [26, Theorem 10] implies that
hold on \(I^2\) with \(\varphi =\int W_{f,g}^{1,0}\) and \(\psi =\int W_{F,G}^{1,0}\). Thus, using that \(W_{f,g}^{1,0}=\gamma W_{F,G}^{1,0}\), we get \(\varphi =\gamma \psi \). This result implies that \(A_\varphi =A_\psi \) is satisfied on \(I^2\). Therefore, assertion (viii) is valid.
The implications (viii) \(\Rightarrow \) (ix) and (ix) \(\Rightarrow \) (i) are obvious. Finally, the equivalence of (vii) and (ix) is a consequence of [26, Corollary 9]. \(\square \)
4.2 Equality of Cauchy means
The second main result of this section is a rephrased form of results of Losonczi [17, 18] who characterized the equality of Cauchy means and established the canonical case and 32 further cases for the equality of these means. The results of Losonczi can easily be deduced from condition (vi) of the next theorem. There, 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 \(6\times 6=36\) subcases which then reduce to the cases considered by Losonczi.
Theorem 16
Let \((f,g),(F,G)\in \mathscr {C}_6(I)\) and let \(\mu \) denote the Lebesgue measure restricted to [0, 1]. Then the following assertions are equivalent:

(i)
The means \(M_{f,g;\mu }\) and \(M_{F,G;\mu }\) are equal on \(I^2\).

(ii)
The means \(M_{f,g;\mu }\) and \(M_{F,G;\mu }\) are equal near the diagonal of \(I^2\).

(iii)
For all \(x\in I\), the equalities in (28) are satisfied.

(iv)
\(\Phi _{f,g}=\Phi _{F,G}\) and there exist constants \(\alpha ,\beta \in \mathbb {R}\) such that
$$\begin{aligned}&\Psi _{f,g}=\alpha \big (W_{f,g}^{1,0}\big )^{\frac{2}{3}} +\tfrac{1}{3}\Phi '\tfrac{2}{9}\Phi ^2\quad \text{ and }\quad \nonumber \\&\Psi _{F,G}=\beta \big (W_{f,g}^{1,0}\big )^{\frac{2}{3}}+\tfrac{1}{3}\Phi '\tfrac{2}{9}\Phi ^2 \end{aligned}$$(39)hold on I.

(v)
Either \((f,g)\sim (F,G)\) or there exist real constants \(a,b,c,A,B,C,\gamma \) such that
$$\begin{aligned}&af^2+bfg+cg^2=\big (W_{f,g}^{1,0}\big )^{\frac{2}{3}} \quad \text{ and }\\&AF^2+BFG+CG^2=\big (W_{F,G}^{1,0}\big )^{\frac{2}{3}} \end{aligned}$$and \(W_{F,G}^{1,0}=\gamma W_{f,g}^{1,0}\).

(vi)
Either \((f,g)\sim (F,G)\) 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 \(\gamma \) and \(\delta \) such that
$$\begin{aligned}&g=\bigg (\frac{f}{g}\bigg )'\bigg (\frac{1}{\sqrt{P^3}}\circ \frac{f}{g}\bigg ), \quad G=\bigg (\frac{F}{G}\bigg )'\bigg (\frac{1}{\sqrt{Q^3}}\circ \frac{F}{G}\bigg ), \quad \text{ and }\nonumber \\&\quad \bigg (\int \frac{1}{Q}\bigg )\circ \frac{F}{G}\quad =\gamma ^{\frac{1}{3}}\bigg (\int \frac{1}{P}\bigg )\circ \frac{f}{g}+\delta . \end{aligned}$$(40) 
(vii)
Either \((f,g)\sim (F,G)\) or there exist a strictly monotone differentiable function \(\varphi :I\rightarrow \mathbb {R}\) and real constants \(\alpha \) and \(\beta \) such that
$$\begin{aligned} \qquad (f,g)\sim (\varphi '\cdot S_\alpha \!\circ \!\varphi ,\varphi '\cdot C_\alpha \!\circ \!\varphi ) \quad \text{ and }\quad (F,G)\sim (\varphi '\cdot S_\beta \!\circ \!\varphi ,\varphi '\cdot C_\beta \!\circ \!\varphi ). \end{aligned}$$ 
(viii)
Either \((f,g)\sim (F,G)\) or \(M_{f,g;\mu }=A_\varphi =M_{F,G;\mu }\) holds on \(I^2\) with \(\varphi :=\int \big (W_{f,g}^{1,0}\big )^{\frac{1}{3}}\).

(ix)
Either \((f,g)\sim (F,G)\) or there exists a strictly monotone function \(\varphi :I\rightarrow \mathbb {R}\) such that \(M_{f,g;\mu }=A_\varphi =M_{F,G;\mu }\) holds on \(I^2\).
Proof
The implication (i) \(\Rightarrow \) (ii) is clear. The implication (ii) \(\Rightarrow \) (iii) is a direct consequence of Lemma 7. Assume now that assertion (iii) is valid. Using that \(\mu \) is the Lebesgue measure restricted to [0, 1], it is easily seen that
Consequently, the conditions \(\mu _2>0=\mu _3\), \(\mu _4\ne 3\mu _2^2\) and \(6\mu _6\mu _2^2\mu _6\mu _45\mu _4^2\mu _2=0\) are valid and using (23), we get \(p=\frac{2}{3}\). Hence the second alternative of Corollary 14 is applicable. Thus \(\Phi _{f,g}=\Phi _{F,G}\) holds and the equalities in (35) reduce to (39), which completes the proof of assertion (iv).
To prove the implication (iv) \(\Rightarrow \) (v), assume that (iv) holds for some constants \(\alpha ,\beta \in \mathbb {R}\). If \(\alpha =\beta \), then Lemma 2 implies that \((f,g)\sim (F,G)\). Now consider the case when \(\alpha \ne \beta \). The existence of some \(\gamma \) such that the identity \(W_{F,G}^{1,0}=\gamma W_{f,g}^{1,0}\) is valid is a direct consequence of the integration of the equality \(\Phi _{f,g}=\Phi _{F,G}\). The equalities in (39) show that the expressions
are constants. Therefore, applying implication (vii) \(\Rightarrow \) (iv) of [21, Theorem 7], we conclude that there exist real constants a, b, c, A, B, C that validate our assertion (v).
Assume now that (v) holds for some constants \(a,b,c,A,B,C,\gamma \) and define
Then, dividing the equalities in (36) 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 (40). Furthermore, using (41), we have that
and similarly,
Applying the equality \(W_{F,G}^{1,0}=\gamma W_{f,g}^{1,0}\), after integration we obtain that the third equality in (40) is also valid for some real constant \(\delta \). This completes the proof of the implication (v) \(\Rightarrow \) (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.
To prove the implication (v) \(\Rightarrow \) (viii), assume that (v) holds for some constants \(a,b,c,A,B,C,\gamma \in \mathbb {R}\). The implication (iv) \(\Rightarrow \) (vi) of [21, Theorem 7] is equivalent to that
hold on \(I^2\) with \(\varphi =\int \Big (W_{f,g}^{1,0}\Big )^{\frac{1}{3}}\) and \(\psi =\int \Big (W_{F,G}^{1,0}\Big )^{\frac{1}{3}}\). Thus, using that \(W_{f,g}^{1,0}=\gamma W_{F,G}^{1,0}\), we get \(\varphi =\gamma ^{\frac{1}{3}}\psi \). This equality yields that \(A_\varphi =A_\psi \) is satisfied on \(I^2\). Therefore, assertion (viii) is valid.
The implications (viii) \(\Rightarrow \) (ix) and (ix) \(\Rightarrow \) (i) are obvious. Finally, the equivalence of the assertions (vii) and (ix) is a consequence of the main result of the paper [12]. \(\square \)
4.3 Conclusion and open problems
We have to stress that the different assertions of Theorem 15 and Theorem 16 require different orders of regularity. Obviously, assertions (i), (ii), (vi), (vii), (ix) make sense in the regularity class \(\mathscr {C}_0(I)\). For (v) and (viii) one has to take the unknown functions from \(\mathscr {C}_1(I)\). Finally, assertions (iv) and (iii) require the regularity class \(\mathscr {C}_2(I)\) and \(\mathscr {C}_6(I)\), 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 \(\mathscr {C}_0(I)\). For the implications (ii) \(\Rightarrow \) (iii) \(\Rightarrow \) (iv), we need \(\mathscr {C}_6(I)\). The proof of the implication (iv) \(\Rightarrow \) (v) requires \(\mathscr {C}_2(I)\), while the equivalence of assertions (v) and (vi), and the implications (v) \(\Rightarrow \) (viii) \(\Rightarrow \) (ix) can be verified in the regularity class \(\mathscr {C}_1(I)\).
Based on the above observations, we can formulate the following three open problems.

1.
Provided that (f, g), \((F,G)\in \mathscr {C}_0(I)\) and \(\mu :=\frac{\delta _0+\delta _1}{2}\) (resp. \(\mu \) is the Lebesgue measure on [0, 1]), are assertions (i), (ii), (vi), (vii), and (ix) of Theorem 15 (resp. Theorem 16) equivalent to each other?

2.
Provided that (f, g), \((F,G)\in \mathscr {C}_1(I)\) and \(\mu :=\frac{\delta _0+\delta _1}{2}\) (resp. \(\mu \) is the Lebesgue measure on [0, 1]), are assertions (i), (ii), (v), (vi), (vii), (viii), and (ix) of Theorem 15 (resp. Theorem 16) equivalent to each other?

3.
Provided that (f, g), \((F,G)\in \mathscr {C}_2(I)\) and \(\mu :=\frac{\delta _0+\delta _1}{2}\) (resp. \(\mu \) is the Lebesgue measure on [0, 1]), are assertions (i), (ii), (iv), (v), (vi), (vii), (viii), and (ix) of Theorem 15 (resp. Theorem 16) equivalent to each other?
References
Aczél, J.: The notion of mean values. Norske Vid. Selsk. Forh. Trondhjem 19(23), 83–86 (1947)
Aczél, J.: On mean values. Bull. Am. Math. Soc. 54, 392–400 (1948)
Aczél, J.: On mean values and operations defined for two variables. Norske Vid. Selsk. Forh. Trondhjem 20(10), 37–40 (1948)
Aczél, J.: On the theory of means. Colloq. Math. 4, 33–55 (1956)
Aczél, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989), With applications to mathematics, information theory and to the natural and social sciences
Bajraktarević, M.: Sur une équation fonctionnelle aux valeurs moyennes. Glasnik Mat.Fiz. Astronom. Društvo Mat. Fiz. Hrvatske Ser. II 13, 243–248 (1958)
Bajraktarević, M.: Sur une généralisation des moyennes quasilinéaires. Publ. Inst. Math. (Beograd) (N.S.) 3(17), 69–76 (1963)
Daróczy, Z., Maksa, Gy., Páles, Zs.: On twovariable means with variable weights. Aequationes Math. 67(12), 154–159 (2004)
de Finetti, B.: Sul concetto di media. Giornale dell’ Instituto, Italiano degli Attuarii 2, 369–396 (1931)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge, (1934) (first edition), (1952) (second edition)
Kiss, T., Páles, Zs.: On a functional equation related to twovariable weighted quasiarithmetic means. J. Differ. Equ. Appl. 24(1), 107–126 (2018)
Kiss, T., Páles, Zs.: On a functional equation related to twovariable Cauchy means. Math. Inequal. Appl. 22(4), 1099–1122 (2019)
Kolmogorov, A.N.: Sur la notion de la moyenne. Rend. Accad. dei Lincei (6) 12, 388–391 (1930)
Losonczi, L.: Equality of two variable weighted means: reduction to differential equations. Aequationes Math. 58(3), 223–241 (1999)
Losonczi, L.: Equality of Cauchy mean values. Publ. Math. Debrecen 57, 217–230 (2000)
Losonczi, L.: Homogeneous Cauchy mean values. In: Daróczy, Z., Páles, Zs. (eds.) Functional Equations—Results and Advances. Advances in Mathematics, vol. 3, pp. 209–218. Kluwer Academic Publishers, Dordrecht (2002)
Losonczi, L.: Equality of two variable Cauchy mean values. Aequationes Math. 65(1–2), 61–81 (2003)
Losonczi, L.: Equality of two variable means revisited. Aequationes Math. 71(3), 228–245 (2006)
Losonczi, L., Páles, Zs.: Comparison of means generated by two functions and a measure. J. Math. Anal. Appl. 345(1), 135–146 (2008)
Losonczi, L., Páles, Zs.: Equality of twovariable functional means generated by different measures. Aequationes Math. 81(12), 31–53 (2011)
Lovas, R. L., Páles, Zs., Zakaria, A.: Characterization of the equality of Cauchy means to quasiarithmetic means. J. Math. Anal. Appl. 484(1) , Art.No. 123700 (2020)
Münnich, Á., Maksa, Gy., Mokken, R. J.: Collective judgement: combining individual value judgements. Math. Social Sci. 37(3), 211–233 (1999)
Nagumo, M.: Über eine Klasse der Mittelwerte. Jpn. J. Math. 7, 71–79 (1930)
Páles, Zs.: On the equality of quasiarithmetic and Lagrangian means. J. Math. Anal. Appl. 382(1), 86–96 (2011)
Páles, Zs., Zakaria, A.: Equality and homogeneity of generalized integral means. Acta Math. Hungar. 160(2), 412–443 (2020)
Páles, Zs., Zakaria, A.: On the equality of Bajraktarević means to quasiarithmetic means. Result. Math. 75(1), Art.No. 19 (2020)
Funding
Open access funding provided by University of Debrecen.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the 80th birthday of Professor Ludwig Reich.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of the second author was supported by the K134191 NKFIH Grant and the 20192.1.11TÉT201900049, the EFOP3.6.116201600022 and the EFOP3.6.216201700015 projects. The last two projects are cofinanced by the European Union and the European Social Fund.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Losonczi, L., Páles, Z. & Zakaria, A. On the equality of twovariable general functional means. Aequat. Math. 95, 1011–1036 (2021). https://doi.org/10.1007/s0001002000755w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s0001002000755w