Comparison of Gini means with fixed number of variables
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910565498093568 |
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| author | Grünwald, Richárd Páles, Zsolt |
| author_facet | Grünwald, Richárd Páles, Zsolt |
| contents | In this paper, we consider the global comparison problem of Gini means with fixed number of variables on a subinterval $I$ of $\mathbb{R}_+$, i.e., the following inequality
\begin{align}\tag{$\star$}\label{ggcabs}
G_{r,s}^{[n]}(x_1,\dots,x_n)
\leq G_{p,q}^{[n]}(x_1,\dots,x_n),
\end{align}
where $n\in\mathbb{N},n\geq2$ is fixed, $(p,q),(r,s)\in\mathbb{R}^2$ and $x_1,\dots,x_n\in I$.
Given a nonempty subinterval $I$ of $\mathbb{R}_+$ and $n\in\mathbb{N}$, we introduce the relations
\[
Γ_n(I):=\{((r,s),(p,q))\in\mathbb{R}^2\times\mathbb{R}^2\mid \eqref{ggcabs}\mbox{ holds for all } x_1,\dots,x_n\in I\},\qquad
Γ_\infty(I):=\bigcap_{n=1}^\inftyΓ_n(I).
\]
In the paper, we investigate the properties of these sets and their dependence on $n$ and on the interval $I$ and we establish a characterizations of these sets via a constrained minimum problem by using a variant of the Lagrange multiplier rule. We also formulate two open problems at the end of the paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07658 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comparison of Gini means with fixed number of variables Grünwald, Richárd Páles, Zsolt Classical Analysis and ODEs 26E60, 26D15, 39B62 In this paper, we consider the global comparison problem of Gini means with fixed number of variables on a subinterval $I$ of $\mathbb{R}_+$, i.e., the following inequality \begin{align}\tag{$\star$}\label{ggcabs} G_{r,s}^{[n]}(x_1,\dots,x_n) \leq G_{p,q}^{[n]}(x_1,\dots,x_n), \end{align} where $n\in\mathbb{N},n\geq2$ is fixed, $(p,q),(r,s)\in\mathbb{R}^2$ and $x_1,\dots,x_n\in I$. Given a nonempty subinterval $I$ of $\mathbb{R}_+$ and $n\in\mathbb{N}$, we introduce the relations \[ Γ_n(I):=\{((r,s),(p,q))\in\mathbb{R}^2\times\mathbb{R}^2\mid \eqref{ggcabs}\mbox{ holds for all } x_1,\dots,x_n\in I\},\qquad Γ_\infty(I):=\bigcap_{n=1}^\inftyΓ_n(I). \] In the paper, we investigate the properties of these sets and their dependence on $n$ and on the interval $I$ and we establish a characterizations of these sets via a constrained minimum problem by using a variant of the Lagrange multiplier rule. We also formulate two open problems at the end of the paper. |
| title | Comparison of Gini means with fixed number of variables |
| topic | Classical Analysis and ODEs 26E60, 26D15, 39B62 |
| url | https://arxiv.org/abs/2408.07658 |