On the invariance of the arithmetic mean with respect to generalized Bajraktarević means
type: Item
instance of: scholarly article
title: On the invariance of the arithmetic mean with respect to generalized Bajraktarević means
author: Richárd Grünwald
author: Zsolt Páles
published in: Acta Mathematica Hungarica
publication date: 2022-06-30T00:00:00Z
review text: 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…
reviewed by: Eszter Gselmann
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 7551271
zbMATH Keywords: quasi-arithmetic mean
zbMATH Keywords: Bajraktarević mean
zbMATH Keywords: invariance equation
MaRDI profile type: Publication
full work available at URL: s10474-022-01230-5
OpenAlex ID: W4224984725
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DOI: 10.1007/S10474-022-01230-5
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description: scientific article; zbMATH DE number 7551271