On the equality problem of two-variable Bajraktarević means under first-order differentiability assumptions

Abstract

The equality problem of two-variable Bajraktarević means can be expressed as the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{-1}\bigg (\frac{f(x)+f(y)}{g(x)+g(y)}\bigg ) =\left( \frac{h}{k}\right) ^{-1}\bigg (\frac{h(x)+h(y)}{k(x)+k(y)}\bigg )\qquad (x,y\in I), \end{aligned}$$ f g - 1 ( f ( x ) + f ( y ) g ( x ) + g ( y ) ) = h k - 1 ( h ( x ) + h ( y ) k ( x ) + k ( y ) ) ( x , y ∈ I ) , where I is a nonempty open real interval, $$f,g,h,k:I\rightarrow {\mathbb {R}}$$ f , g , h , k : I → R are continuous functions, g, k are positive and f/g, h/k are strictly monotone. This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. Additional and new characterizations of this equality problem have been found recently by Losonczi, Páles and Zakaria under the same regularity assumptions in 2021. In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability.

On the Equality of Weighted Bajratarević Means to Quasi-Arithmetic Means

“…Several new characterizations of the equality of two-variable Bajraktarević means have been obtained by Losonczi, Páles and Zakaria in [4] under the same regularity assumptions. Recently, Páles and Zakaria [5] obtained the same conclusion under only first-order differentiability. Grünwald and Páles [6] considered the equality problem of generalized Bajraktarević means.…”