The optimization for the inequalities of power means

Abstract

Let M [t] n (a) be the tth power mean of a sequence a of positive real numbers, where a = (a 1 ,a 2 ,...,a n ),n ≥ 2, and α,λ ∈ R m ++ ,m ≥ 2, m j=1 λ j = 1,min{α} ≤ θ ≤ max {α}. In this paper, we will state the important background and meaning of the inequalityn (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.Copyright © 2006 J. Wen and W.-L. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Symbols and introductionWe will use some symbols in the well-known monographs [1,5,13]:is a finite set, and it is not empty.Recall that the definitions of the tth power mean and Hardy mean of order r for a sequence a = (a 1 ,...,a n ) (n ≥ 2) are, respectively,, if 0< |t| < +∞,n (a) = n √ a 1 a 2 ··· a n , if t = 0, Hindawi Publishing Corporation

Cyclic inequalities involving Cater cyclic function

“…In this paper, we will study the lower with upper bounds of the Cater cyclic function and establish two Cater-type cyclic inequalities, and display the applications of the dimension reduction method [2,3,5,7] in the inequality theory.…”

Some new results for power means

“…Due to the importance of power mean in modern mathematics, it has also been given considerable attention by mathematicians. Many remarkable results for power mean have been presented in the literature (see, for example, [2,3,4,5,6,11,14,15,16,17] and the references cited therein).…”

Closed balls for interpolating quasi-polynomials

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