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arXiv · 1703.02354

On the local and global comparison of generalized Bajraktarević means

Zsolt Páles·Amr Zakaria
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Abstract

Given two continuous functions $f,g:I\to\mathbb{R}$ such that $g$ is positive and $f/g$ is strictly monotone, a measurable space $(T,A)$, a measurable family of $d$-variable means $m: I^d\times T\to I$, and a probability measure $μ$ on the measurable sets $A$, the $d$-variable mean $M_{f,g,m;μ}:I^d\to I$ is defined by $$ M_{f,g,m;μ}(\pmb{x}) :=\left(\frac{f}{g}\right)^{-1}\left( \frac{\int_T f\big(m(x_1,\dots,x_d,t)\big) dμ(t)} {\int_T g\big(m(x_1,\dots,x_d,t)\big) dμ(t)}\right) \qquad(\pmb{x}=(x_1,\dots,x_d)\in I^d). $$ The aim of this paper is to study the local and global comparison problem of these means, i.e., to find conditions for the generating functions $(f,g)$ and $(h,k)$, for the families of means $m$ and $n$, and for the measures $μ,ν$ such that the comparison inequality $$ M_{f,g,m;μ}(\pmb{x})\leq M_{h,k,n;ν}(\pmb{x}) \qquad(\pmb{x}\in I^d) $$ be satisfied.

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Zsolt Páles, Amr Zakaria. 2017-03-07. On the local and global comparison of generalized Bajraktarević means. https://doi.org/10.1016/j.jmaa.2017.05.073

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