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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 850215, 21 pages doi:10.1155/2010/850215 Research Article Jensen Type Inequalities Involving Homogeneous Polynomials Jia-Jin Wen 1 and Zhi-Hua Zhang 2 1 College of Mathematics and Information Science, Chengdu University, Sichuan 610106, China 2 Department of Mathematics, Shili Senior High School in Zixing, Chenzhou, Hunan 423400, China Correspondence should be addressed to Jia-Jin Wen, wenjiajin623@163.com Received 4 November 2009; Revised 25 January 2010; Accepted 8 February 2010 Academic Editor: Soo Hak Sung Copyright q 2010 J J. Wen and Z H. Zhang. 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. By means of algebraic, analytical and majorization theories, and under the proper hypotheses, we establish several Jensen type inequalities involving γth homogeneous polynomials as follows:  m k1 w k fX k /fI n  ≤ f  m k1 w k X γ k /fI n  1/γ ,  m k1 w k fX k /fN n  ≤ f  m k1 w k X γ k / fN n  1/γ ,and  m k1 w k f ∗ X k  ≤ f ∗   m k1 w k X k , and display their applications. 1. Introduction The following notation and hypotheses in 1–4 will be used throughout the paper: x   x 1 ,x 2 , ,x n  † ,αα 1 ,α 2 , ,α n  † ,w  w 1 ,w 2 , ,w m  † , X k   x k,1 ,x k,2 , ,x k,n  † , N  { 0, 1, 2, ,n, } ,n∈ N,n≥ 2, R   −∞, ∞  , R n  0, ∞ n , R n  0, ∞ n , Ω n  { x ∈ R n  | 0 ≤ x 1 ≤ x 2 ≤··· ≤ x n } . 1.1 Also let P γ  x   ⎧ ⎨ ⎩   α,σ  ∈B γ × S n λ  α, σ  n  j1 x α j σ  j        λ : B γ × S n → R ⎫ ⎬ ⎭ \ { 0 } , P  γ  x   ⎧ ⎨ ⎩   α,σ  ∈B γ × S n λ  α, σ  n  j1 x α j σ  j        λ : B γ × S n −→  0, ∞  ⎫ ⎬ ⎭ \ { 0 } , 2 Journal of Inequalities and Applications P γ  x   ⎧ ⎨ ⎩  α∈B γ λ  α  n! per  x α i j        λ : B γ → R ⎫ ⎬ ⎭ \ { 0 } , P  γ  x   ⎧ ⎨ ⎩  α∈B γ λ  α  n! per  x α i j        λ : B γ →  0, ∞  ⎫ ⎬ ⎭ \ { 0 } , 1.2 where  x α i j    x α i j  n×n  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x α 1 1 x α 1 2 x α 1 3 ··· x α 1 n x α 2 1 x α 2 2 x α 2 3 ··· x α 2 n . . . . . . . . . . . . . . . x α n 1 x α n 2 x α n 3 ··· x α n n ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ n×n , 1.3 B γ is a nonempty and finite subset of  α ∈ R n       n  i1 α i  γ,γ∈  0, ∞   , 1.4 and the permanent of n × n matrix A a i,j  n×n is given by see 2, 4 per A   σ∈S n n  j1 a j,σj ; 1.5 here, the sum extends over all elements σ of the nth symmetric group S n . If f ∈ P γ x, then f is called γth homogeneous polynomial; if f ∈ P γ x, then f is called γth homogeneous symmetric polynomial see 3. The famous Jensen inequality can be stated as follows: if f : I → R is a convex function, then for any x ∈ I n we have 1 n n  k1 f  x k  ≥ f  1 n n  k1 x k  . 1.6 A large number of generalizations and applications of the inequality 1.6 had been obtained in 1 and 5–8. An interesting generalization of 1.6 was given by Chen et al., in 8:LetB γ ⊂ N n and f ∈ P  γ x.IfX k ∈ R n  with 1 ≤ k ≤ m and 0 ≤ X 1 ≤ X 2 ≤ ··· ≤ X m , then we have the following Jensen type inequality: 1 m m  k1 f  X k  ≥ f  1 m m  k1 X k  . 1.7 Journal of Inequalities and Applications 3 In this paper, by means of algebraic, analytical, and majorization theories, and under the proper hypotheses, we will establish several Jensen type inequalities involving γth homogeneous polynomials and display their applications. 2. Jensen Type Inequalities Involving Homogeneous Polynomials In this section, we will use the following notation see 1, 4, 9: Q    q p     p ∈ N \ { 0 } ,q∈ N \ { 0 }  ,I n   1, 1, ,1  † ,N n   1, 2, ,n  † , x γ   x γ 1 ,x γ 2 , ,x γ n  † ,φ  x  φx 1 ,φx 2 , ,φx n  † ,A  x   1 n n  i1 x i , Δx   Δx 1 , Δx 2 , ,Δx n  †   x 1 ,x 2 − x 1 ,x 3 − x 2 , ,x n − x n−1  † . 2.1 2.1. A Jensen Type Inequality Involving Homogeneous Polynomials We begin a Jensen type inequality involving homogeneous polynomials as follows. Theorem 2.1. Let f ∈ P  γ x.IfX k ∈ R n  with 1 ≤ k ≤ m and w ∈ R m  ,then  m k1 w k f  X k  f  I n  ≤ ⎡ ⎢ ⎣ f   m k1 w k X γ k  f  I n  ⎤ ⎥ ⎦ 1/γ . 2.2 The equality holds in 2.2 if there exists t ∈ 0, ∞, such that X 1  X 2  ··· X m  tI n . Lemma 2.2. (H ¨ older’s inequality, see [1, 10]). Let a i,k ∈ 0, ∞,q i ∈ 0, ∞ with 1 ≤ i ≤ n and 1 ≤ k ≤ m.If  n i1 q i ≤ 1,then 1 m m  k1 n  i1 a q i i,k ≤ n  i1  1 m m  k1 a i,k  q i . 2.3 The equality in 2.3 holds if a i,1  a i,2  ··· a i,m for 1 ≤ i ≤ n. Lemma 2.3. (Power mean inequality, see [1, 10–11]). Let x ∈ R n  ,μ ∈ R n  and  n i1 μ i  1.If γ ∈ 1, ∞,then n  i1 μ i x γ i ≥  n  i1 μ i x i  γ . 2.4 The inequality is reversed for γ ∈ 0, 1. The equality in 2.4 holds if and only if γ  1,or x 1  x 2  ··· x n . 4 Journal of Inequalities and Applications Lemma 2.4. Let gx, α  n j1 x α j σj and σ ∈ S n .Ifα ∈B γ and X k ∈ R n  with 1 ≤ k ≤ m,then g  m  k1 X γ k ,α  ≥  m  k1 g  X k ,α   γ . 2.5 The equality in 2.5 holds if α 1, 0, ,0 † , or there exists t ∈ 0, ∞, such that X 1  X 2  ··· X m  tI n . Proof. According to α ∈B γ ,  n j1 α j /γ1 ≤ 1 and Lemmas 2.2-2.3,wegetthat g  1 m m  k1 X γ k ,α   n  j1  1 m m  k1 x γ k,σ  j   α j  ⎡ ⎣ n  j1  1 m m  k1 x γ k,σ  j   α j /γ ⎤ ⎦ γ ≥ ⎡ ⎣ 1 m m  k1 n  j1 x α j k,σ  j  ⎤ ⎦ γ   1 m m  k1 g  X k ,α   γ . 2.6 From g  1 m m  k1 X γ k ,α   1 m γ g  m  k1 X γ k ,α  , 2.7 we deduce to the inequality 2.5. Lemma 2.4 is proved. Proof of Theorem 2.1. First of all, we assume that w  I m . According to γ ∈ 1, ∞, fI n   α,σ∈B γ ×S n λα, σ and Lemmas 2.3-2.4 ,wefindthat f   m k1 X γ k  f  I n    α,σ∈B γ ×S n λ  α, σ  f  I n  g  m  k1 X γ k ,α  ≥   α,σ  ∈B γ ×S n λ  α, σ  f  I n   m  k1 g  X k ,α   γ ≥ ⎡ ⎣   α,σ  ∈B γ ×S n λ  α, σ  f  I n  m  k1 g  X k ,α  ⎤ ⎦ γ    m k1 f  X k  f  I n   γ . 2.8 That is, the inequality 2.2 holds. Journal of Inequalities and Applications 5 Secondly, for some of w k with 1 ≤ k ≤ m satisfing w k /  1, we have the following cases. 1 If w ∈ N m , then the inequality 2.2 holds from the above proof. 2 If w ∈ Q m  , then there exists N ∈ N \{0} that satisfies Nw ∈ N m .Bytheresult in 1,weobtainthat m  k1 Nw k f  X k  f  I n  ≤ ⎡ ⎢ ⎢ ⎢ ⎣ f  m  k1 Nw k X γ k  f  I n  ⎤ ⎥ ⎥ ⎥ ⎦ 1/γ ⇐⇒ m  k1 w k f  X k  f  I n  ≤ ⎡ ⎢ ⎢ ⎢ ⎣ f  m  k1 w k X γ k  f  I n  ⎤ ⎥ ⎥ ⎥ ⎦ 1/γ , 2.9 which implies that inequality 2.2 is also true. 3 If w ∈ R m  , then there exist sequences {w i k } ∞ i1 , such that w i k ∈ Q   1 ≤ i<∞  , lim i →∞ w i k  w k  1 ≤ k ≤ m  . 2.10 We get by the case in 2 that  m k1 w i k f  X k  f  I n  ≤ ⎡ ⎢ ⎣ f   m k1 w i k X γ k  f  I n  ⎤ ⎥ ⎦ 1/γ , 2.11 and taking i →∞in 2.11, we can get the inequality 2.2. T he proof of Theorem 2.1 is thus completed. 2.2. Jensen Type Inequalities Involving Difference Substitution Exchange the ith row and jth row in nth unit matrix E, then this matrix, written Ei, j,is called nth exchange matrix. If E 1 , E 2 , ,E p are nth exchange matrixes, then the n × n matrix D n  E p E p−1 ···E 1 E 0 Δ n is called nth difference matrix, and the substitution x  D n y is difference substitution, where p ∈ N, E 0  E,and Δ n  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 10··· 0 11··· 0 . . . . . . . . . . . . 11··· 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ n×n . 2.12 6 Journal of Inequalities and Applications Let f ∈ P γ x.IffD n y ∈ P  γ y is true for any difference matrix D n , then fx ≥ 0 for any x ∈ R n  see 11, and the homogeneous polynomial f is called positive semidefinite with difference substitution. If we let D n  { D n | D n be nth difference matrix } , P ∗ γ  x    f  x  ∈ P γ  x  |B γ ⊂ N n ,f  D n y  ∈ P  γ  y  , ∀D n ∈D n  , 2.13 then D n is a finite set and the count of elements of D n is |D n |  n!, and γ ∈ N. We have the following Jensen type inequality involving homogeneous polynomials and difference substitution. Theorem 2.5. Let f ∈ P ∗ γ x.Ifw ∈ R m  , and X k ∈ Ω n with 1 ≤ k ≤ m,then  m k1 w k f  X k  f  N n  ≤ ⎡ ⎢ ⎣ f   m k1 w k X γ k  f  N n  ⎤ ⎥ ⎦ 1/γ . 2.14 The equality holds in 2.14 if there exists t ∈ 0, ∞, such that X 1  X 2  ···  X m  tI n , and fI n 0. Lemma 2.6. (Jensen’s inequality, see [12]). For any x ∈ R n  and γ ∈ 1, ∞, we have  n  k1 x k  γ ≥ n  k1 x γ k . 2.15 The equality in 2.33 holds if and only if γ  1, or at least n − 1 numbers equal zero among the set {x 1 ,x 2 , ,x n }. Lemma 2.7. If γ ∈ 1, ∞ and x ∈ Ω n , then for the difference substitution x Δ n y, one has the following double inequality: 0 ≤ y γ ≤ Δx γ . 2.16 The equality y γ Δx γ holds if and only if γ  1,orx 1  x 2  ··· x n−1  0,orx 1  x 2  ··· x n . Proof. From x ∈ Ω n , it is easy to know that y Δ −1 n x Δx ∈ R n  .Byγ ∈ 1, ∞ and Lemma 2.6, we find that 0 ≤ y γ 1  x γ 1 ≤ x γ 1 , 0 ≤ y γ 2   x 2 − x 1  γ ≤ x γ 2 − x γ 1 , . . . 0 ≤ y γ n   x n − x n−1  γ ≤ x γ n − x γ n−1 . 2.17 This shows that the double inequality 2.16 holds. Journal of Inequalities and Applications 7 Proof of Theorem 2.5. Consider the difference substitution X k Δ n Y k . Since X k ∈ Ω n , Y k  Δ −1 n X k ΔX k ∈ R n  with 1 ≤ k ≤ m.Fromf ∈ P ∗ γ x, we have that fD n y ∈ P  γ y, for all D n ∈D n . Hence, f  Δ n y  ∈ P  γ  y  . 2.18 According to Theorem 2.1,weobtainthat  m k1 w k f  Δ n Y k  f  Δ n I n  ≤ ⎡ ⎢ ⎣ f  Δ n  m k1 w k Y γ k  f  Δ n I n  ⎤ ⎥ ⎦ 1/γ  ⎡ ⎢ ⎣ f   m k1 w k Δ n Y γ k  f  N n  ⎤ ⎥ ⎦ 1/γ . 2.19 In view of Y k ∈ R n  and with Lemma 2.7, we have 0 ≤ Y γ k ≤ ΔX γ k ,k 1, 2, ,m. 2.20 By noting that fΔ n y ∈ P  γ y, it implies that f  m k1 w k Δ n Y γ k  is increasing with respect to Y γ k .Thus, f  m  k1 w k Δ n Y γ k  ≤ f  m  k1 w k Δ n ΔX γ k   f  m  k1 w k X γ k  . 2.21 Therefore,  m k1 w k f  X k  f  N n    m k1 w k f  Δ n Y k  f  Δ n I n  ≤ ⎡ ⎢ ⎣ f   m k1 w k Δ n Y γ k  f  N n  ⎤ ⎥ ⎦ 1/γ ≤ ⎡ ⎢ ⎣ f   m k1 w k X γ k  f  N n  ⎤ ⎥ ⎦ 1/γ . 2.22 This evidently completes the proof of Theorem 2.5. As an application of Theorem 2.5, we have the following. Theorem 2.8. Let fxAx γ  − A γ x,γ ∈ N and γ ≥ 2. If w ∈ R m  ,X k ∈ Ω n with 1 ≤ k ≤ m, then the inequality 2.14 holds. The equality holds in 2.14 if there exists t ∈ 0, ∞, such that X 1  X 2  ··· X m  tI n . 8 Journal of Inequalities and Applications Proof. First of all, we prove that f ∈ P ∗ γ x. If the function φ : I → R satisfies the condition that φ  : I → R is continuous, then we have the following identity: A  φ  x   − φ  A  x   1 n 2  1≤i<j≤n   ∇ φ   t 1 x i  t 2 x j   1 − t 1 − t 2  A  x   dt 1 dt 2  x i − x j  2 , 2.23 where x ∈ I n ,φ   t   d 2 φ dt 2 , ∇   t 1 ,t 2  † ∈ R 2  | t 1  t 2 ≤ 1  . 2.24 In fact,  ∇ φ   t 1 x i  t 2 x j   1 − t 1 − t 2  A  x   dt 1 dt 2   1 0 dt 1  1−t 1 0 φ   t 1 x i  t 2 x j   1 − t 1 − t 2  A  x   dt 2  1 x j − A  x   1 0 dt 1  1−t 1 0 φ   t 1 x i  t 2 x j   1 − t 1 − t 2  A  x   d  t 1 x i  t 2 x j   1 − t 1 − t 2  A  x    1 x j − A  x   1 0 dt 1 φ   t 1 x i  t 2 x j   1 − t 1 − t 2  A  x     1−t 1 0  1 x j − A  x   1 0  φ   t 1 x i   1 − t 1  x j  − φ   t 1 x i   1 − t 1  A  x   dt 1  1 x j − A  x   φt 1 x i 1 − t 1 x j  x i − x j − φt 1 x i 1 − t 1 Ax x i − Ax       1 0  1 x j − A  x   φ  x i  − φ  x j  x i − x j − φ  x i  − φ  A  x  x i − A  x    1  x i − x j  x j − A  x    x i − A  x         φ  A  x  A  x  1 φ  x i  x i 1 φ  x j  x j 1        , 2.25 and  1≤i<j≤n   ∇ φ   t 1 x i  t 2 x j   1 − t 1 − t 2  A  x   dt 1 dt 2  x i − x j  2   1≤i<j≤n x i − x j  x j − A  x    x i − A  x         φ  A  x  A  x  1 φ  x i  x i 1 φ  x j  x j 1        Journal of Inequalities and Applications 9  1 2  1≤i,j≤n  1 x j − A  x  − 1 x i − A  x           φ  A  x  A  x  1 φ  x i  x i 1 φ  x j  x j 1          1 2 ⎛ ⎜ ⎜ ⎝  1≤i,j≤n 1 x j − A  x          φ  A  x  A  x  1 φ  x i  x i 1 φ  x j  x j 1         −  1≤i,j≤n 1 x i − A  x          φ  A  x  A  x  1 φ  x i  x i 1 φ  x j  x j 1         ⎞ ⎟ ⎟ ⎠  1 2 ⎛ ⎜ ⎜ ⎝ n  j1 1 x j − A  x  n  i1         φ  A  x  A  x  1 φ  x i  x i 1 φ  x j  x j 1         − n  i1 1 x i − A  x  n  j1         φ  A  x  A  x  1 φ  x i  x i 1 φ  x j  x j 1         ⎞ ⎟ ⎟ ⎠  1 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ n  j1 n x j − A  x            φ  A  x  A  x  1 1 n n  i1 φ  x i  A  x  1 φ  x j  x j 1           − n  i1 n x i − A  x             φ  A  x  A  x  1 φ  x i  x i 1 1 n n  j1 φ  x j  A  x  1            ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠  1 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ n  j1 n x j − A  x              φ  A  x  − 1 n n  i1 φ  x i  00 1 n n  i1 φ  x i  A  x  1 φ  x j  x j 1             − n  i1 n x i − A  x             φ  A  x  A  x  1 φ  x i  x i 1 1 n n  j1 φ  x j  − φ  A  x  00            ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠  n 2 ⎧ ⎨ ⎩ n  j1 −  A  φ  x   − φ  A  x   A  x  − x j  x j − A  x  − n  i1  A  φ  x   − φ  A  x    A  x  − x i  x i − A  x  ⎫ ⎬ ⎭  n 2 ⎧ ⎨ ⎩ n  j1  A  φ  x   − φ  A  x    n  i1  A  φ  x   − φ  A  x   ⎫ ⎬ ⎭  n 2  A  φ  x   − φ  A  x   . 2.26 That is, the identity 2.23 holds. 10 Journal of Inequalities and Applications Setting φ :  0, ∞  −→ R,φ  t   t γ 2.27 in 2.23, we have that f  x   1 n 2  1≤i<j≤n   ∇ γ  γ − 1  t 1 x i  t 2 x j 1 − t 1 − t 2 Ax  γ−2 dt 1 dt 2  x i − x j  2 . 2.28 Since f ∈ P γ x, f ∈ P ∗ γ x if and only if fΔ n y ∈ P  γ y. Consider the difference substitution x Δ n y.From  x i − x j  2   j  ki1 y k  2 ∈ P  2  y  ⇒  x i − x j  2 ∈ P ∗ 2  x  2.29 for arbitrary i, j :1≤ i<j≤ n,itiseasytoseethatf ∈ P ∗ 2 x if γ  2. If γ ≥ 3, then  x i − x j  2 ∈ P ∗ 2  x  ,  t 1 x i  t 2 x j 1 − t 1 − t 2 Ax  γ−2 ∈ P  γ−2  x  ⇒  ∇ γ  γ − 1  t 1 x i  t 2 x j 1 − t 1 − t 2 Ax  γ−2 dt 1 dt 2 ∈ P  γ−2  x  ⇒  ∇ γ  γ − 1  t 1 x i  t 2 x j 1 − t 1 − t 2 Ax  γ−2 dt 1 dt 2 ∈ P ∗ γ−2  x  2.30 for arbitrary i, j :1≤ i<j≤ n. Therefore, we get that f ∈ P ∗ γ x. It follows that the inequality 2.14 holds by using Theorem 2.5. Since fI n 0, the equality holds in 2.14 if there exists t ∈ 0, ∞, such that X 1  X 2  ··· X m  tI n . The proof of Theorem 2.8 is thus completed. Remark 2.9. Theorem 2.8 has significance in the theory of matrices. Let A a i,j  n×n be an n × n positive definite Hermitian matrix and λ 1 , ,λ n its eigenvalues, let diagx be the diagonal matrix with the components of x x 1 ,x 2 , ,x n  † as its diagonal elements, and also let λ λ 1 ,λ 2 , ,λ n  † . Then A  U diagλU ∗ for some unitary matrix U where U ∗ is the conjugate transpose of U and U ∗ U  E, see 9, 13.Ifγ ∈ R, then A γ  U diag  λ γ  U ∗ , tr A  n  i1 a i,i  n  i1 λ i , tr A γ  n  i1 λ γ i . 2.31 [...]... inequality 3.3 in 4 : if X1 , X2 ∈ n Ω , and α ∈ Rn , then we have the following Chebyshev type inequality: per X1 X2 αi j ≥ n! per X1 αi j × n! per X2 αi j n! 3.4 3.1 Jensen Type Inequalities Involving Homogeneous Symmetric Polynomials In this subsection, we first present a Jensen type inequality involving homogeneous symmetric polynomials as follows 1, w ∈ Nm , let Bγ be a control ordered set If... Wen, Inequalities of Jensen- Peˇ ari´ -Svrtan-Fan type, ” Journal of Inequalities in Pure and c c Applied Mathematics, vol 9, no 3, article 74, 8 pages, 2008 8 Y.-X Chen, J.-Y Luo, and J.-K Yang, “A class of Jensen inequalities for homogeneous and symmetric polynomials,” Journal of Sichuan Normal University, vol 30, pp 481–484, 2007 Chinese 9 J.-J Wen, S S Cheng, and C Gao, “Optimal sublinear inequalities. .. 3 Jensen Type Inequalities Involving Homogeneous Symmetric Polynomials In this section, we will also use the following notation see 4, 16 : ex e x1 , e x2 , , e xn † , αl α1 , α2 , , αn l l l m Ωn ∗ † ∈ Bγ , m Xk p k 1 X1 X2 l † m xk,2 , , k 1 l l max α1 , α2 , , αn 1≤l≤N m xk,1 , k 1 {x ∈ Rn | x1 ≤ x2 ≤ · · · ≤ xn }, xk,n k 1 x1,1 x1,2 x1,n , , , x2,1 x2,2 x2,n † , 3.1 , Journal of Inequalities. .. inequalities involving permanents and their applications,” Linear Algebra and Its Applications, vol 422, no 1, pp 295–303, 2007 5 J Peˇ ari´ and D Svrtan, “New refinements of the Jensen inequalities based on samples with c c repetitions,” Journal of Mathematical Analysis and Applications, vol 222, no 2, pp 365–373, 1998 6 Z.-G Xiao, H M Srivastava, and Z.-H Zhang, “Further refinements of the Jensen inequalities. .. 3.7 The inequality 3.6 is also a Chebyshev type inequality involving homogeneous symmetric polynomials 3.3 An Open Problem According to Theorem 3.3, we pose the following open problem Conjecture 3.8 Under the hypotheses of Theorem 3.3, one has ⎡ X1 1 per⎣ n! X2 αi j 1 ⎤ ⎦≤ αi j 1 α X2 j i 1 per X1 per ≤ f X1 ≤ f X2 n i 1 n i 1 p x1,i p x2,i γ/p 3.21 Journal of Inequalities and Applications 21 References... 481–484, 2007 Chinese 9 J.-J Wen, S S Cheng, and C Gao, “Optimal sublinear inequalities involving geometric and power means,” Mathematica Bohemica, vol 134, no 2, pp 133–149, 2009 10 J.-J Wen and W.-L Wang, “The optimization for the inequalities of power means,” Journal of Inequalities and Applications, vol 2006, Article ID 46782, 25 pages, 2006 11 L Yang, Y Feng, and Y Yao, “A class of mechanically... 1045–1058, 2008 15 J.-J Wen and R.-X Zhang, “Two conjectured inequalities involving the sums of inscribed polygons in some circles,” Journal of Shanxi Normal University, vol 30, supplement 1, pp 12–17, 2002 Chinese 16 J Peˇ ari´ , J.-J Wen, W.-L Wang, and T Lu, “A generalization of Maclaurin’s inequalities and its c c applications,” Mathematical Inequalities and Applications, vol 8, no 4, pp 583–598, 2005... increasing with 1 ≤ k ≤ m Then the inequality 2.14 can be rewritten as follows: m k 1 wk Dγ fk ξ Dγ ξ0 ≤ ⎫1/γ γ wk fk ξ ⎬ ⎧ ⎨ Dγ m k 1 ⎩ Dγ ξ0 ⎭ , 2.40 where w ∈ Rm , γ ∈ N, γ ≥ 2 2.3 Applications of Jensen Type Inequalities By 1.7 and the same proving method of Theorem 2.1, we can obtain the following result m k 1 Corollary 2.11 Let Bγ ⊂ Nn , f ∈ P γ x If w ∈ Rm , 0 ≤ X1 ≤ X2 ≤ · · · ≤ Xm , then m wk f... Their Inequalities, Reidel, Dordrecht, The c c Netherlands, 1988 2 H Minc, Permanents, vol 9999 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, Mass, USA, 1978 3 V Timofte, “On the positivity of symmetric polynomial functions I General results,” Journal of Mathematical Analysis and Applications, vol 284, no 1, pp 174–190, 2003 4 J.-J Wen and W.-L Wang, “Chebyshev type inequalities. .. pp 1611–1620, 2007 e 12 W V Jensen, “Sur les fonctions convexes et les in´ galit´ s entre les valeurs moyennes,” Acta e Mathematica, vol 30, no 1, pp 175–193, 1906 13 B Mond and J E Peˇ ari´ , “Generalization of a matrix inequality of Ky Fan,” Journal of Mathematical c c Analysis and Applications, vol 190, no 1, pp 244–247, 1995 14 J.-J Wen and W.-L Wang, Inequalities involving generalized interpolation . hypotheses, we will establish several Jensen type inequalities involving γth homogeneous polynomials and display their applications. 2. Jensen Type Inequalities Involving Homogeneous Polynomials In this. Corporation Journal of Inequalities and Applications Volume 2010, Article ID 850215, 21 pages doi:10.1155/2010/850215 Research Article Jensen Type Inequalities Involving Homogeneous Polynomials Jia-Jin. x 1 ,x 3 − x 2 , ,x n − x n−1  † . 2.1 2.1. A Jensen Type Inequality Involving Homogeneous Polynomials We begin a Jensen type inequality involving homogeneous polynomials as follows. Theorem 2.1.