On the $σ$-balancing property of multivariate generalized quasi-arithmetic means
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866913350442549248 |
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| author | Kiss, Tibor Nagy, Gergő |
| author_facet | Kiss, Tibor Nagy, Gergő |
| contents | The aim of this paper is to characterize the so-called $σ$-balancing property in the class of generalized quasi-arithmetic means. In general, the question is whether those elements of a given family of means that possess this property are quasi-arithmetic.
The first result in the latter direction is due to G. Aumann who showed that a balanced complex mean is necessariliy quasi-arithmetic provided that it is analytic. Then Aumann characterized quasi-arithmetic means among Cauchy means in terms of the balancing property. These results date back to the 1930s. In 2015, Lucio R. Berrone, generalizing balancedness, concluded that a mean having that more general property is quasi-arithmetic if it is symmetric, strict and continuously differentiable. A common feature of these results is that they assume a certain order of differentiability of the mean whether or not it is a natural condition.
In 2020, the balancing property was characterized in the family of generalized quasi-arithmetic means of two variables under only natural conditions, namely continuity and strict monotonicity of their generating functions. Here we extend the corresponding result for multivariate generalized quasi-arithmetic means by relaxing the conditions on the generating functions and considering the more general $σ$-balancing property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08583 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $σ$-balancing property of multivariate generalized quasi-arithmetic means Kiss, Tibor Nagy, Gergő Classical Analysis and ODEs Primary 39B22, Secondary 26E60 The aim of this paper is to characterize the so-called $σ$-balancing property in the class of generalized quasi-arithmetic means. In general, the question is whether those elements of a given family of means that possess this property are quasi-arithmetic. The first result in the latter direction is due to G. Aumann who showed that a balanced complex mean is necessariliy quasi-arithmetic provided that it is analytic. Then Aumann characterized quasi-arithmetic means among Cauchy means in terms of the balancing property. These results date back to the 1930s. In 2015, Lucio R. Berrone, generalizing balancedness, concluded that a mean having that more general property is quasi-arithmetic if it is symmetric, strict and continuously differentiable. A common feature of these results is that they assume a certain order of differentiability of the mean whether or not it is a natural condition. In 2020, the balancing property was characterized in the family of generalized quasi-arithmetic means of two variables under only natural conditions, namely continuity and strict monotonicity of their generating functions. Here we extend the corresponding result for multivariate generalized quasi-arithmetic means by relaxing the conditions on the generating functions and considering the more general $σ$-balancing property. |
| title | On the $σ$-balancing property of multivariate generalized quasi-arithmetic means |
| topic | Classical Analysis and ODEs Primary 39B22, Secondary 26E60 |
| url | https://arxiv.org/abs/2405.08583 |