Abstract
Let Mn[t](a) be the tth power mean of a sequence a of positive real numbers, where a=(a1,a2,…,an),n≥2, and α,λ∈ℝ++m,m≥2,∑j=1mλj=1,min{α}≤θ≤max{α}. In this paper, we will state the important background and meaning
of the inequality ∏j=1m{Mn[αj](a)}λj≤(≥)Mn[θ](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.