Equality and homogeneity of generalized integral means

• Published in: Acta mathematica Hungarica 2020-04, Vol.160 (2), p.412-443
• Main Authors: Páles, Zs, Zakaria, A.
• Format: Article
• Language: English
• Subjects: Continuity (mathematics); Equality; Homogeneity; Mathematics; Mathematics and Statistics
• ISSN: 0236-5294; 1588-2632
• DOI: 10.1007/s10474-019-01012-6

Given two continuous functions f , g : I → 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 × T → I , and a probability measure μ on the measurable sets A , the d -variable mean M f , g , m ; μ : I d → I is defined by M f , g , m ; μ ( x ) : = ( f g ) - 1 ( ∫ T f ( m ( x 1 , … , x d , t ) ) d μ ( t ) ∫ T g ( m ( x 1 , … , x d , t ) ) d μ ( t ) ) ( x = ( x 1 , … , x d ) ∈ 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 ; μ ( x ) = M h , k , m ; μ ( x ) ( x ∈ I d ) and the homogeneity property M f , g , m ; μ ( λ x ) = λ M f , g , m ; μ ( x ) ( λ > 0 , x , λ x ∈ I d ) , respectively, be satisfied.