On the local and global comparison of generalized Bajraktarević means
Zsolt Páles; Amr Zakaria;
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 $$
M_{f,g,m;\mu}(\pmb{x})
:=\left(\frac{f}{g}\right)^{-1}\left(
\frac{\int_T f\big(m(x_1,\dots,x_d,t)\big) d\mu(t)}
{\int_T g\big(m(x_1,\dots,x_d,t)\big) d\mu(t)}\right)
\qquad(\pmb{x}=(x_1,\dots,x_d)\in I^d). $$ 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 $\mu,\nu$ such that the comparison
inequality $$
M_{f,g,m;\mu}(\pmb{x})\leq M_{h,k,n;\nu}(\pmb{x}) \qquad(\pmb{x}\in I^d) $$
be satisfied.
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 $$
M_{f,g,m;\mu}(\pmb{x})
:=\left(\frac{f}{g}\right)^{-1}\left(
\frac{\int_T f\big(m(x_1,\dots,x_d,t)\big) d\mu(t)}
{\int_T g\big(m(x_1,\dots,x_d,t)\big) d\mu(t)}\right)
\qquad(\pmb{x}=(x_1,\dots,x_d)\in I^d). $$ 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 $\mu,\nu$ such that the comparison
inequality $$
M_{f,g,m;\mu}(\pmb{x})\leq M_{h,k,n;\nu}(\pmb{x}) \qquad(\pmb{x}\in I^d) $$
be satisfied.
Other data
Title | On the local and global comparison of generalized Bajraktarević means | Authors | Zsolt Páles; Amr Zakaria | Keywords | Mathematics - Classical Analysis and ODEs; Mathematics - Classical Analysis and ODEs; Primary 26D10, 26D15, Secondary 26B25, 39B72, 41A50 | Issue Date | 7-Mar-2017 | Journal | J. Math. Anal. Appl. 455 (2017), 792-815 | ISSN | 0022247X | DOI | 10.1016/j.jmaa.2017.05.073 |
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