Equality and homogeneity of generalized integral means

Given two continuous functions $f,g:I\to \mathbb{R}$ such that $g$ is positive and $f\mathrm{/}g$ 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 $\mu$ on the measurable sets $A$, the $d$-variable mean $M_{f,g,m;\mu }:I^d\to I$ is defined by $$M_{f,g,m;\mu }(x):={\left(\frac{f}{g}\right)}^{-1}\left(\frac{{\int }_{T}f(m(x_1,\dots ,x_d,t))d\mu (t)}{{\int }_{T}g(m(x_1,\dots ,x_d,t))d\mu (t)}\right)(x=(x_1,\dots ,x_d)\in I^d)\mathrm{.}$$ The aim of this paper is to solve the equality and homogeneity problems of these means, i.e., to find conditions for the generating functions $(f,g)$ and $(h,k)$, for the family of means $m$, and for the measure $\mu$ such that the equality $$M_{f,g,m;\mu }(x)=M_{h,k,m;\mu }(x)(x\in I^d)$$ and the homogeneity property $$M_{f,g,m;\mu }(\lambda x)=\lambda M_{f,g,m;\mu }(x)(\lambda >0,x,\lambda x\in I^d),$$ respectively, be satisfied.