On the equality of two-variable general functional means

Abstract

AbstractGiven two functions $$f,g:I\rightarrow \mathbb {R}$$ f , g : I → R and a probability measure $$\mu $$ μ on the Borel subsets of [0, 1], the two-variable mean $$M_{f,g;\mu }:I^2\rightarrow I$$ M f , g ; μ : I 2 → I is defined by $$\begin{aligned} M_{f,g;\mu }(x,y) :=\bigg (\frac{f}{g}\bigg )^{-1}\left( \frac{\int _0^1 f\big (tx+(1-t)y\big )d\mu (t)}{\int _0^1 g\big (tx+(1-t)y\big )d\mu (t)}\right) \quad (x,y\in I). \end{aligned}$$ M f , g ; μ ( x , y ) : = ( f g ) - 1 ∫ 0 1 f ( t x + ( 1 - t ) y ) d μ ( t ) ∫ 0 1 g ( t x + ( 1 - t ) y ) d μ ( t ) ( x , y ∈ 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 $$\mu $$ μ , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which $$\begin{aligned} M_{f,g;\mu }(x,y)=M_{F,G;\mu }(x,y) \quad (x,y\in I) \end{aligned}$$ M f , g ; μ ( x , y ) = M F , G ; μ ( x , y ) ( x , y ∈ I ) holds. Under at most sixth-order differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarević means and of Cauchy means.