Local and global comparison of generalized Bajraktarević means

Richárd Grünwald; Zsolt Páles

paper

Local and global comparison of generalized Bajraktarević means

arXiv:2112.11871

Abstract

The purpose of this paper is to investigate the local and global comparison of two $n$ -variable generalized Bajraktarević means, i.e., to establish necessary as well as sufficient conditions in terms of the unknown functions $f, g, p_{1}, \dots, p_{n}, q_{1}, \dots, q_{n} : I \to R$ for the comparison inequality $f^{- 1} (\frac{p _{1} ( x _{1} ) f ( x _{1} ) + \dots + p _{n} ( x _{n} ) f ( x _{n} )}{p _{1} ( x _{1} ) + \dots + p _{n} ( x _{n} )}) \leq g^{- 1} (\frac{q _{1} ( x _{1} ) g ( x _{1} ) + \dots + q _{n} ( x _{n} ) g ( x _{n} )}{q _{1} ( x _{1} ) + \dots + q _{n} ( x _{n} )})$ 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 $p_{1}, \dots, p_{n}, q_{1}, \dots, q_{n} : I \to R_{+}$ 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 $x_{1}, \dots, x_{n} \in I$ .