Equality and homogeneity of generalized integral means

Abstract

Given two continuous functions $f,g \colon I \to\mathbb{R}$ such that g is positive and f/g is strictly monotone, a measurable space $(T,\mathcal{A})$, a measurable family of d-variable means $m: I^{d} \times T \to I$, and a probability measure μ on the measurable sets $\mathcal{A}$, the d-variable mean $M_{f,g,m;\mu} \colon I^{d} \to I$ is defined by

$$M_{f,g,m;\mu}({\bf x}) :=\Bigl(\frac{f}{g}\Bigr)^{-1}\biggl( \frac{\int_{T} f(m(x_{1},\ldots,x_{d},t)){\rm d}\mu(t)} {\int_{T} g(m(x_{1},\ldots,x_{d},t)){\rm d}\mu(t)}\biggr) \quad ({\bf x} =(x_{1},\ldots,x_{d})\in I^{d}).$$

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 μ such that the equality

$$M_{f,g,m;\mu}( {\bf x} )=M_{h,k,m;\mu}( {\bf x} ) \quad ( {\bf x} \in I^{d})$$

and the homogeneity property

$$M_{f,g,m;\mu}(\lambda {\bf x} ) = \lambda M_{f,g,m;\mu}( {\bf x} ) \quad (\lambda>0,\, {\bf x} ,\lambda {\bf x} \in I^{d}), $$

respectively, be satisfied.