Characterizations of the equality of two-variable generalized quasiarithmetic means

广义拟算术平均的相等性与比较

Equality and comparison of generalized quasiarithmetic means

本研究拓展拟算术平均定义为采用严格单调生成函数的广义形式，建立其性质并与标准形式对比，解决了其可比性与相等性问题，还给出了依赖区间或变量数决定能否表示为该类平均的示例。

Characterization of the equality of Cauchy means to quasiarithmetic means

The main result of this paper provides six necessary and sufficient conditions under various regularity assumptions for a so-called Cauchy mean to be identical to a two-variable quasiarithmetic mean. One of these conditions says that a Cauchy mean is quasiarithmetic if and only if the range of its generating functions is covered by a nondegenerate conic section.

由不连续严格单调函数生成的广义拟算术平均的等式问题

Equality problem for generalized quasiarithmetic means generated by discontinuous strictly monotonic functions

研究由不连续严格单调函数生成的广义拟算术平均的等式问题，给出两个充分条件得类似结论，举例说明无额外条件时生成函数非仿射变换，其余情况考虑函数逆得出缩放因子在连续区域一致的结论。

On the equality of two-variable general functional means

Given two functions $f,g:I\to\mathbf{R}$ and a probability measure $μ$ on the Borel subsets of $[0,1]$, the two-variable mean $M_{f,g;μ}:I^2\to I$ is defined by $$ M_{f,g;μ}(x,y) :=\bigg(\frac{f}{g}\bigg)^{-1}\left( \frac{\int_0^1 f\big(tx+(1-t)y\big)dμ(t)} {\int_0^1 g\big(tx+(1-t)y\big)dμ(t)}\right) \qquad(x,y\in I). $$ This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević means. The aim of this paper is, for a fixed probability measure $μ$, to study their equality problem, i.e., to characterize those pairs of functions $(f,g)$ and $(F,G)$ such that $$ M_{f,g;μ}(x,y…

Equality and homogeneity of generalized integral means

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 $μ$ on the measurable sets $A$, the $d$-variable mean $M_{f,g,m;μ}:I^d\to I$ is defined by $$ M_{f,g,m;μ}(\pmb{x}) :=\left(\frac{f}{g}\right)^{-1}\left( \frac{\int_T f\big(m(x_1,\dots,x_d,t)\big)\,dμ(t)} {\int_T g\big(m(x_1,\dots,x_d,t)\big)\,dμ(t)}\right) \qquad(\pmb{x}=(x_1,\dots,x_d)\in I^d). $$ The aim of this paper is to solve the equality and homogeneity problems…

On the equality problem of generalized Bajraktarević means

The purpose of this paper is to investigate the equality problem of generalized Bajraktarević means, i.e., to solve the functional equation \begin{equation}\label{E0}\tag{*} f^{(-1)}\bigg(\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)}\bigg)=g^{(-1)}\bigg(\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)}\bigg), \end{equation} which holds for all $x=(x_1,\dots,x_n)\in I^n$, where $n\geq 2$, $I$ is a nonempty open real interval, the unknown functions $f,g:I\to\mathbb{R}$ are strictly monotone, $f^{(-1)}$ and $g^{(-1)}$ denote their generalized left inverses,…