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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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