Local and global comparison of generalized Bajraktarevi\'c means

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 $x_1,\dots,x_n\in I$.