Lower estimation of the difference among quasi-arithmetic means

Quasi-arithmetic means are defined for every continuous, strictly monotone function $f \colon U \rightarrow \mathbb{R}$, ($U$ -- an interval). For an $n$-tuple $a \in U^n$ with corresponding vector of weights $w=(w_1,\dots,w_n)$ ($w_i>0$, $\sum w_i=1$) it equals $f^{-1}\left( \sum_{i=1}^{n} w_i f(a_i)\right)$. In 1960s Cargo and Shisha defined a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013--16 we proved that, having two quasi-arithmetic means, we can majorized distance between them in terms of Arrow-Pratt index $f''/f'$. In this paper we are going to proof that this operator can be also used to establish certain lower boundaries of this distance.