engleski

The optimal power mean bounds for two convex combinations of A-G-H means

For $p \in \R$, let $M_p(a, b)$ denote the usual power mean of order $p$ of positive real numbers $a$ and $b$, and let $A = M_1$, $G=M_0$ and $H=M_{; ; ; -1}; ; ; $. We prove that the inequalities $M_0(a, b) \leq \frac{; ; ; 1}; ; ; {; ; ; 3}; ; ; [A(a, b) + G(a, b) + H(a, b)] \leq M_{; ; ; \frac{; ; ; \ln 2}; ; ; {; ; ; \ln 6}; ; ; }; ; ; (a, b)$ and $M_{; ; ; -\frac{; ; ; 1}; ; ; {; ; ; 6}; ; ; }; ; ; (a, b) \leq \frac{; ; ; 1}; ; ; {; ; ; 2}; ; ; [He(a, b) + H(a, b)] \leq M_{; ; ; \frac{; ; ; \ln 2}; ; ; {; ; ; \ln 6}; ; ; }; ; ; (a, b)$ hold for all positive real numbers $a$ and $b$, with strict inequality for $a \neq b$, and that the orders of power means involved are optimal.

arithmetic mean; geometric mean; harmonic mean; power mean; Heronian mean; sharp inequality

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