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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.

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