1 Introduction
2 Main results
3 Applications to special means
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The arithmetic mean$$ A=A(a,b)=\frac{a+b}{2},\quad a,b\in \mathbb{R}\text{ with } a,b>0; $$
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The geometric mean$$ G=G(a,b)=\sqrt{ab},\quad a,b\in \mathbb{R}\text{ with } a,b>0; $$
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The harmonic mean$$ H=H(a,b)=\frac{2ab}{a+b},\quad a,b\in \mathbb{R}\backslash \{0\}; $$
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The identric mean$$ I=I(a,b)=\textstyle\begin{cases} a & \text{if $a=b$,} \\ \frac{1}{e} ( \frac{b^{b}}{a^{a}} ) ^{\frac{1}{b-a}} & \text{if $a\neq b, a,b>0$;} \end{cases} $$
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The logarithmic mean$$ L=L(a,b)=\textstyle\begin{cases} a & \text{if $a=b$,} \\ {\frac{b-a}{\ln b-\ln a}} & \text{if $a\neq b$;} \end{cases} $$
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Generalized logarithmic mean$$ L_{n}(a,b)=\textstyle\begin{cases} a & \text{if $a=b$,} \\ { [\frac{b^{n+1}-a^{n+1}}{(n+1)(b-a)} ]^{\frac{1}{n}}} & \text{if $a\neq b, n\in \mathbb{Z}\backslash {\{-1,0\}}, a,b>0$.} \end{cases} $$