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\begin{document}$${I\subset \mathbb {R}}$$\end{document} be a nonvoid open interval. A function \documentclass[12pt]{minimal}
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\begin{document}$${K:I^2\to I}$$\end{document} is called an M-conjugate mean if there exists \documentclass[12pt]{minimal}
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\begin{document}$${(p,q)\in [0,1]^2}$$\end{document} and a continuous strictly monotone real valued function \documentclass[12pt]{minimal}
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\begin{document}$${\varphi}$$\end{document} on I such that\documentclass[12pt]{minimal}
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\begin{document}$$K(x,y)=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(M(x,y)))=:M_ \varphi^{(p,q)}(x,y)$$\end{document}holds for all \documentclass[12pt]{minimal}
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\begin{document}$${x,y\in I}$$\end{document}. In this paper, we investigate the equality and comparison problem in the class of M-conjugate means, in the case when
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\begin{document}$$M(x,y):=\min\{x,y\}\quad (x,y\in I)$$\end{document}.