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On the invariance equation for two-variable weighted nonsymmetric Bajraktarević means
- Source :
- Aequationes mathematicae. 93:37-57
- Publication Year :
- 2018
- Publisher :
- Springer Science and Business Media LLC, 2018.
-
Abstract
- The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevic means, i.e., to solve the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{\!\!-1}\!\!\left( \frac{tf(x)+sf(y)}{tg(x)+sg(y)}\right) +\left( \frac{h}{k}\right) ^{\!\!-1}\!\!\left( \frac{sh(x)+th(y)}{sk(x)+tk(y)}\right) =x+y \qquad (x,y\in I), \end{aligned}$$ where $$f,g,h,k:I\rightarrow \mathbb {R}$$ are unknown continuous functions such that g, k are nowhere zero on I, the ratio functions f / g, h / k are strictly monotone on I, and $$t,s\in \mathbb {R}_+$$ are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that $$f,g,h,k:I\rightarrow \mathbb {R}$$ are four times continuously differentiable.
- Subjects :
- Weight function
39B12, 39.35, 26E60
Applied Mathematics
General Mathematics
010102 general mathematics
Hyperbolic function
Zero (complex analysis)
010103 numerical & computational mathematics
01 natural sciences
Combinatorics
Monotone polygon
Természettudományok
Mathematics - Classical Analysis and ODEs
Functional equation
Discrete Mathematics and Combinatorics
Matematika- és számítástudományok
0101 mathematics
Mathematics
Variable (mathematics)
Subjects
Details
- ISSN :
- 14208903 and 00019054
- Volume :
- 93
- Database :
- OpenAIRE
- Journal :
- Aequationes mathematicae
- Accession number :
- edsair.doi.dedup.....b1219cbd00dc2db4813d24e9f208f7a0