Abstract
Given 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.
An Improved Alternating CQ Algorithm for Solving Split Equality Problems
