, Volume 84, Issue 1-2, pp 77-90
Date: 25 Jan 2012

The equality problem in the class of conjugate means

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Abstract

Let \({I\subset\mathbb{R}}\) be a nonempty open interval and let \({L:I^2\to I}\) be a fixed strict mean. A function \({M:I^2\to I}\) is said to be an L-conjugate mean on I if there exist \({p,q\in{]}0,1]}\) and a strictly monotone and continuous function φ such that

$$M(x,y):=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(L(x,y)))=:L_\varphi^{(p,q)}(x,y),$$
for all \({x,y\in I}\) . Here L(x, y) is a fixed quasi-arithmetic mean. We will solve the equality problem in this class of means.