Equality of two variable means revisited

Summary.

Let I be an interval,

$$ f,g:I \to \mathbb{R} $$

be given continuous functions such that g(x)≠ 0 for x ∈I and h(x) : =  f(x)/g(x) (x ∈I) 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. M_{f, 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.