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Sharp inequalities related to one-parameter mean and Gini mean. (English) Zbl 1257.26028
Summary: In the present paper, we answer the question: For $$\alpha +\beta \in (0,1)$$, what are the greatest values $$p,s_1$$ and the least values $$q,s_2$$ such that the inequalities $J_p(a,b) \leqslant A^\alpha (a,b)G^\beta (a,b)H^{1 - \alpha - \beta} (a,b) \leqslant J_q(a,b)$ and $G_{s_1},_{1}(a,b) \leqslant A^\alpha (a,b)G^\beta (a,b)H^{1 - \alpha - \beta} (a,b) \leqslant G_{s_2,1}(a,b)$ hold for all $$a,b > 0$$ with $$a \neq b$$ ? where $$J_p(a,b), A(a,b), G(a,b), H(a,b)$$ and $$G_{s,1}(a,b)$$ are the one-parameter mean, arithmetic mean, geometric mean, harmonic mean and Gini mean for two positive numbers $$a$$ and $$b$$, respectively.

##### MSC:
 26E60 Means 26D07 Inequalities involving other types of functions 26D99 Inequalities in real analysis
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