On the local and global comparison of generalized Bajraktarević means

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 $\mu$ on the measurable sets $A$, the $d$-variable mean $M_{f,g,m;\mu }:I^d\to I$ is defined by $$M_{f,g,m;\mu }(x):={\left(\frac{f}{g}\right)}^{-1}\left(\frac{{\int }_{T}f(m(x_1,\dots ,x_d,t))d\mu (t)}{{\int }_{T}g(m(x_1,\dots ,x_d,t))d\mu (t)}\right)(x=(x_1,\dots ,x_d)\in I^d).$$ 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 $\mu ,\nu$ such that the comparison inequality $$M_{f,g,m;\mu }(x)\le M_{h,k,n;\nu }(x)(x\in I^d)$$ be satisfied.

Cited by in corpus (2)

• On the invariance equation for two-variable weighted nonsymmetric Bajraktarević means
• Equality and homogeneity of generalized integral means