, 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.

This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK 81402 (first and second author) and OTKA “Mobility” call HUMAN-MB08A-84581 (first author).