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Proof of one optimal inequality for generalized logarithmic, arithmetic, and geometric means. (English) Zbl 1210.26033
The main theorem is an answer on two open problems given by B.-Y. Long and Y.-M. Chu [J. Inequal. Appl. 2010, Article ID 806825 (2010; Zbl 1187.26015)].
Theorem: Let $$\alpha \in (0,1/2)\cup (1/2,1)$$, $$a\not= b$$, $$a>0$$, $$b>0$$. Let $$p(\alpha)$$ be a solution of $\frac 1p \ln (1+p)+\ln (\frac{\alpha}{2})=0$ in $$(-1,1)$$. Then, if $$\alpha \in (0,1/2)$$, then $\alpha A(a,b)+(1-\alpha)G(a,b) <L_p(a,b)$ for $$p\geq p(\alpha)$$ and $$p(\alpha)$$ is the best constant, and if $$\alpha \in (1/2,1)$$, then $\alpha A(a,b)+(1-\alpha)G(a,b) >L_p(a,b)$ for $$p\leq p(\alpha)$$ and $$p(\alpha)$$ is the best constant.
$$A(a,b), G(a,b)$$ and $$L_p(a,b)$$ are notations for an arithmetic mean, a geometric mean and a generalized logarithmic mean respectively.

##### MSC:
 26E60 Means 26D99 Inequalities in real analysis
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##### References:
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