The optimization for the inequalities of power means. (English) Zbl 1133.26324

Summary: Let \(M_n[t](a)\) be the \(t\)-th power mean of a sequence a of positive real numbers, where \(a=(a_1,a_2,\dots, a_n)\), \(n\geq 2\), and \(\alpha,\lambda\in\mathbb{R}_{++}\), \(m\geq 2,\sum_{j=1}^m\lambda_j=1\), \(\min\{\alpha\}\leq \theta\leq \max\{\alpha\}\). In this paper, we will state the important background and meaning of the inequality \(\prod_{j=1}^m \{M_n^{[\alpha_j]}(a)\}^{\lambda_j}\leq (\geq) M_n^{[\theta]}(a)\); a necessary and sufficient condition and another interesting sufficient condition that the foregoing inequality holds are obtained; an open problem posed by Wang et al. in 2004 is solved and generalized; a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also obtained. Our methods used are the procedure of descending dimension and theory of majorization; and apply techniques of mathematical analysis and permanents in algebra.


26E60 Means
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