The optimization for the inequalities of power means
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a of positive real numbers, where Let Mn[t] (a) be the tth power mean of a sequence m a = (a1 ,a2 ,...,an ),n ≥ 2, and α,λ ∈ Rm ++ ,m ≥ 2, j =1 λ j = 1,min {α} ≤ θ ≤ max {α}. In this paper, we will state the important background and meaning of the inequality m
λj [α j ] j =1 {Mn (a)}
[θ]
≤ (≥ )Mn (a); a necessary and sufficient condition and another in-
teresting 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. 1. Symbols and introduction We will use some symbols in the well-known monographs [1, 5, 13]: An = a = (a1 ,...,an ), aθ = (aθ1 ,...,aθn ), In = (1,...,1), On = (0,...,0), α = (α1 ,...,αm ); min{α} = min{α1 ,...,αm }; max{α} = max{α1 ,...,αm }, λ = (λ1 ,...,λm ); Rn = {a : ai ∈ R,1 ≤ i ≤ n}; Rn+ = {a : ai ≥ 0, 1 ≤ i ≤ n}, Rn++ = {a : ai > 0, 1 ≤ i ≤ n}, Zn+ = {a | ai ≥ 0, ai is a integer, i = 1,2,...,n}, (0,1]n = {a : 0 < ai ≤ 1, 1 ≤ i ≤ n}, d ∈ R, Bd ⊂ {α : α ∈ Rm , α1 + · · · + αm = d }, Bd 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 = (a1 ,...,an ) (n ≥ 2) are, respectively,
Mn[t] (a) = √
1 t · a n i =1 i n
1/t
Mn[t] (a) = n a1 a2 · · · an , Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 46782, Pages 1–25 DOI 10.1155/JIA/2006/46782
,
if 0 < |t | < +∞, if t = 0,
2
The optimization for the inequalities of power means
Hn (a;r) =
1 r j · ai n! i1 ,...,in j =1 j n
√
Hn (a;r) = n a1 a2 · · · an ,
1/(r1 +···+rn )
if r1 + · · · + rn > 0,
,
if ri = 0, i = 1,...,n, (1.1)
where t ∈ R, r ∈ Rn+ , a ∈ Rn++ . And 1 rj · a , n! i1 ,...,in j =1 i j n
hn (a;r) =
a ∈ Rn++ , r ∈ Rn ,
(1.2)
is called Hardy function, where i1 ,...,in is the total permutation of 1, ...,n. Definition 1.1. Let α ∈ Rn , let λα be a function of α, λα ∈ R, x ∈ Rn++ . Then the function f (x) = α∈Bd λα hn (x;α) is called the generalized homogeneous symmetrical polynomial of n variables and degree d. When Bd ⊂ Z+n , f (x) is called the homogeneous symmetrical polynomial of n variables and degree d, simply, homogeneous symmetrical polynomial (see [24, page 431]). Definition 1.2. Let ai j be the complex numbers, i, j = 1,2,...,n, and let the matrix A = (ai j )n×n be an n × n matrix. Then the permanent (of order n) of A is a function of matrix, written perA, it is defined by perA =
a1σ1 a2σ2 · · · anσn ,
(1.3)
σ
where the summation extends over all one-to-one functions from 1,..
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