aequationes mathematicae

, Volume 71, Issue 3, pp 228–245

Equality of two variable means revisited

Original Paper

Summary.

Let I be an interval,
$$ f,g:I \to \mathbb{R} $$
be given continuous functions such that g(x)≠ 0 for xI and h(x) : =  f(x)/g(x) (xI) is strictly monotonic (thus invertible) on I. Let further μ be an increasing non-constant function on [0, 1]. We consider the function
$$ M_{{f,g,\mu }} (x,y): = h^{{ - 1}} {\left( {\frac{{{\int\limits_0^1 {f(tx + (1 - t)y)d\mu (t)} }}} {{{\int\limits_0^1 {g(tx + (1 - t)y)d\mu (t)} }}}} \right)}(x,y \in I) $$
where the integrals are Riemann – Stieltjes ones. From the mean value theorem it follows that
$$ \min \{ x,y\} \leqslant M_{{f,g,\mu }} (x,y) \leqslant \max \{ x,y\} $$
i.e. Mf, g, μ is a mean on I. With suitable choice of μ we can get from it the quasi-arithmetic mean weighted by a weight function, and also the Cauchy or difference mean.

The equality problem for these two classes of means has been solved in [22], [28]. The aim of this paper is twofold. First, we solve these equality problems in a unified way, second we get rid of the inconvenient conditions (vanishing or not vanishing of some functions) posed in the previous papers.

Mathematics Subject Classification (2000).

Primary 39B22 

Keywords.

Divided differences mean value functional equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Institute of MathematicsDebrecen UniversityDebrecenHungary

Personalised recommendations