On the invariance of the arithmetic mean with respect to generalized Bajraktarević means
Given a nonempty open interval \(I\subset \mathbb{R}\), a strictly monotone continuous function \(f \colon I \to \mathbb{R}\) and a pair of positive valued functions \(p = (p_1, p_2) \colon I \to \mathbb{R}^{2}_{+}\), the \(2\)-variable generalized Bajraktarević mean \(A_{f,p} \colon I^{2} \to I\) is given by the following formula \[ A_{f, p}= f^{-1}\left(\frac{p_{1}(x)f(x)+p_{2}(x)f(y)}{p_{1}(x)+p_{2}(y)}\right) \qquad \left(x, y\in I\right). \] The main purpose of this paper is to solve the invariance equation of two \(2\)-variable generalized Bajraktarević means with respect to the arithmetic mean, i.e., to solve the functional equation \[(*)\;\;\; A_{f, p}(x, y)+A_{g, q}(x, y)=x+y \qquad \left(x, y\in I\right), \] for the unknown strictly monotone and continuous functions \(f, g \colon I \to \mathbb{R}\) and for the also unknown functions \(p, q \colon I \to \mathbb{R}^{2}_{+}\).\par While solving invariance equation \((\ast)\), the authors assume that the unknown functions \(f\) and \(g\) of the class \(\mathcal{C}^{4}(I)\), while \(p, q\in \mathcal{C}^{2}(I)\). In such a way, they are able to prove and to use formulas for the partial derivatives of generalized Bajraktarević means at diagonal points of \(I^{2}\). In connection to this, several formula were proved in their former paper [Aequationes Math. 94, No. 4, 651--677 (2020; Zbl 1447.39013)] under tight regularity assumptions, here they also calculate the fourth-order partial derivative \(\partial^{2}_{x} \partial^{2}_{y} A_{f, p}\).\par The main result of the paper is Theorem 10 that reads as follows. \par Let \(f, g \colon I \to \mathbb{R}\) be four times continuously differentiable functions on \(I\) with non-vanishing first derivatives. Let \(p = (p_1, p_2 ) \colon I\to \mathbb{R}^{2}_{+}\) be a twice continuously differentiable function and \(q = (q_1, q_2) \colon I\to \mathbb{R}^{2}_{+}\). Assume that the set \[P = \left\{x \in I \, \vert \, p_1(x) = p_2 (x) \right\}\] is nowhere dense in \(I\). Then the following assertions are equivalent to each other.\par (i) The invariance equation \((\ast)\) holds for every \((x, y) \in I^{2}\). \par (ii) There exists an open set \(U\subset I^{2}\) containing the diagonal diag\((I^2)\) such that the invariance equation \((*)\) holds for all \((x, y) \in U\).\par (iii) The function \(q = (q_1, q_2 ) \colon I \to \mathbb{R}^{2}_{+}\) is twice continuously differentiable and the system of equations \[\partial_{x}A_{f, p}+\partial_{x}A_{g, q}=1 \] \[ \partial_{x}\partial_{y}A_{f, p}+\partial_{x}\partial_{y}A_{g, q}=0 \] \[ \partial_{x}^{2}\partial_{y}A_{f, p}+\partial_{x}^{2}\partial_{y}A_{g, q}=0\] \[ \partial_{x}\partial_{y}^{2}A_{f, p}+\partial_{x}^{2}\partial_{y}^{2}A_{g, q}=0\] holds on diag\((I^{2})\). \par (iv) There exists a constant \(\gamma \in \mathbb{R}\), there exist solutions \(u, v, w, z \colon I \to \mathbb{R}\) of the second-order linear differential equation \(F''= \gamma F\) such that \(v > 0\) and \(z > 0\) holds on \(I\) and \(\{u, v\}\) and \(\{w, z\}\) are linearly independent such that \[f= \frac{u}{v}, \qquad g= \frac{w}{z}, \qquad \text{and} \qquad p_{1}q_{1}= p_{2}q_{2}= vz\] hold.
scientific article; zbMATH DE number 7551271
scientific article