Local and global comparison of generalized Bajraktarevi\'c means

TL;DR

This paper establishes necessary and sufficient conditions for comparing generalized Bajraktarević means, providing a comprehensive analysis of local and global inequalities based on the properties of the involved functions.

Contribution

It offers new criteria for the comparison of generalized Bajraktarević means, including conditions involving derivatives and ratios of weight functions, extending previous results.

Findings

Derived necessary and sufficient conditions for mean comparison

Established global comparison criteria under differentiability assumptions

Provided explicit inequalities involving function ratios and derivatives

Abstract

The purpose of this paper is to investigate the local and global comparison of two $n$-variable generalized Bajraktarevi\'c means, i.e., to establish necessary as well as sufficient conditions in terms of the unknown functions $f,g,p_1,\dots,p_n,q_1,\dots,q_n:I\to\mathbb{R}$ for the comparison inequality $$ f^{-1}\bigg(\frac{p_1(x_1)f(x_1)+\cdots+p_n(x_n)f(x_n)}{p_1(x_1)+\cdots+p_n(x_n)}\bigg)\leq g^{-1}\bigg(\frac{q_1(x_1)g(x_1)+\cdots+q_n(x_n)g(x_n)}{q_1(x_1)+\cdots +q_n(x_n)}\bigg) $$ in local and global sense. Here $I$ is a nonempty open real interval, $x_1,\dots,x_n\in I$, and $f,g$ is assumed to be continuous, strictly monotone and $p_1,\dots,p_n,q_1,\dots,q_n:I\to\mathbb{R}_+$ are positive valued. Concerning the global comparison problem, the main result of the paper states that if $f,g$ are differentiable functions with nonvanishing first derivatives and, for all $i\in\{1,\dots,n\}$, $$ \frac{p_i}{p_0}=\frac{q_i}{q_0} \qquad\mbox{and}\qquad \frac{p_0(x)(f(x)-f(y))}{p_0(y)f'(y)} \leq\frac{q_0(x)(g(x)-g(y))}{q_0(y)g'(y)}\qquad(x,y\in I) $$ are satisfied (where $p_0:=p_1+\dots+p_n$ and $q_0:=q_1+\dots+q_n$), then the above comparison inequality holds for all $x_1,\dots,x_n\in I$.