, Volume 81, Issue 3, pp 201-208
Date: 27 Apr 2011

On the equality and comparison problem of a class of mean values

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Abstract

Let \({I\subset \mathbb {R}}\) be a nonvoid open interval. A function \({K:I^2\to I}\) is called an M-conjugate mean if there exists \({(p,q)\in [0,1]^2}\) and a continuous strictly monotone real valued function \({\varphi}\) on I such that $$K(x,y)=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(M(x,y)))=:M_ \varphi^{(p,q)}(x,y)$$ holds for all \({x,y\in I}\) . In this paper, we investigate the equality and comparison problem in the class of M-conjugate means, in the case when $$M(x,y):=\min\{x,y\}\quad (x,y\in I)$$ .

This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK 81402.
Dedicated to Professor Walter Benz on the occasion of his 80th birthday.