S-GEOMETRIC CONVEXITY OF A FUNCTION INVOLVING MACLAURIN’S ELEMENTARY

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S-Geometric Convexity Xiao-ming Zhang vol. 8, iss. 2, art. 51, 2007

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S-GEOMETRIC CONVEXITY OF A FUNCTION INVOLVING MACLAURIN’S ELEMENTARY

SYMMETRIC MEAN

XIAO-MING ZHANG

Zhejiang Haining TV University, Haining, Zhejiang,

314400, P. R. China.

EMail:zjzxm79@sohu.com Received: 23 February, 2007

Accepted: 27 April, 2007

Communicated by: P.S. Bullen

2000 AMS Sub. Class.: Primary 26D15.

Key words: Geometrically convex function, S-geometrically convex function, Inequality, Maclaurin-Inequality, Logarithm majorization.

Abstract: Letxi > 0, i = 1,2, . . . , n, x = (x1, x2, . . . , xn), thekth elementary symmetric function of x is defined as En(x, k) = P

1≤i₁<···<i_k≤n k

Q

j=1

xi_j with 1 ≤ k ≤ n, the kth elementary symmetric mean is defined asPn(x, k) = n

k

−1

En(x, k)_k¹

, and the functionfis defined asf(x) =Pn(x, k−1)− Pn(x, k). The paper proves thatf is a S-geometrically convex function. The result generalizes the well-known Maclaurin-Inequality.

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1. Introduction

Throughout the paper we assumeRⁿbe then-dimensional Euclidean Space, Rⁿ+ ={(x₁, x₂, . . . , x_n), x_i >0, i= 1,2, . . . , n},

and

e^x = (e^x¹, e^x², . . . , e^xⁿ), x^c = (x^c₁, x^c₂, . . . , x^c_n), lnx= (lnx₁,lnx₂, . . . ,lnx_n), x·y= (x₁y₁, x₂y₂, . . . , x_ny_n),

wherec ∈ R, and x = (x₁, x₂, . . . , x_n) ∈ Rⁿ, y = (y₁, y₂, . . . , y_n) ∈ Rⁿ. And if n≥2,x= (x₁, x₂, . . . , x_n)∈Rⁿ+. They are defined respectively by

A_n(x) = x₁+x₂+· · ·+x_n

n , G_n(a) = √ⁿ

x₁x₂· · ·x_n.

Thekth elementary symmetric function ofx, kth elementary symmetric mean, and functionf are defined respectively as

E_n(x, k) = X

1≤i1<···<i_k≤n k

Y

j=1

x_i_j, (1≤k ≤n)

P_n(x, k) = n

k −1

E_n(x, k) ¹_k

, (1≤k≤n) f(x) = P_n(x, k−1)−P_n(x, k), (2≤k≤n) with ⁿ_k

= _k!(n−k)!^n! .

The following theorem is true by [1].

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Theorem 1.1 (Maclaurin-Inequality).

A_n(x) =P_n(x,1) (1.1)

≥P_n(x,2)≥ · · · ≥P_n(x, n−1)≥P_n(x, n) =G_n(x). E_n(x, n)

E_n(x, n−1) < E_n(x, n−1) E_n(x, n−2) (1.2)

<· · ·< E_n(x,3)

En(x,2) < E_n(x,2)

En(x,1) < E_n(x,1).

References [5], [4], [2], [7], [8], [9], [10] and [6] give the definitions of n dimensional geometrically convex functions, S-geometrically convex functions and logarithm majorization, and a large number of results have been obtained. Since many functions have geometric convexity or geometric concavity, research into geometrically convex functions make sense. For a comprehensive list of recent results on geometrically convex functions, see the book [10] and the papers [5], [4], [2] [7]

[8], [9] and [6] where further results are given.

The main aim of this paper is to prove the following theorem.

Theorem 1.2. Letn = 2,orn≥3,2≤k−1≤n−1, andf(x) = P_n(x, k−1)− P_n(x, k), thenf is aS−geometrically convex function.

The result generalizes the Maclaurin-Inequality.

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2. Relative Definition and a Lemma

Lemma 2.1 ([3]). LetH ⊆Rⁿbe a symmetric convex set with a nonempty interior, φ : H → Rbe continuously differentiable on the interior of H and continuous on H. Necessary and sufficient conditions forφto be S-convex(concave) onHare that φis symmetric onH, and

(x₁−x₂) ∂φ

∂x₁ − ∂φ

∂x₂

≥(≤) 0, for allxin the interior ofH.

Definition 2.1 ([9], [10, p. 89], [6]). Letx ∈ Rⁿ+, y ∈Rⁿ+, x_[1], x_[2], . . . , x_[n]

and y_[1], y_[2], . . . , y_[n]

be the decreasing queues of(x₁, x₂, . . . , x_n)and(y₁, y₂, . . . , y_n) respectively. We say (x₁, x₂, . . . , x_n) logarithm majorizes (y₁, y₂, . . . , y_n), denote lnxlnyif











k

Q

i=1

x_i ≥

k

Q

i=1

y_i, k = 1,2, . . . , n−1,

n

Q

i=1

x_i =

n

Q

i=1

y_i. Remark 1. Ifxlogarithm majorizesy, then









 ln

_k Q

i=1

x_i

≥ln _k

Q

i=1

y_i

, k= 1,2, . . . , n−1,

ln _n

Q

i=1

x_i

= ln _n

Q

i=1

y_i

.

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









k

P

i=1

lnx_i ≥

k

P

i=1

lny_i, k= 1,2, . . . , n−1,

n

P

i=1

lnx_i =

n

P

i=1

lny_i.

So we can denotelnxlnyin the Definition2.1.

Lemma 2.2 ([10, p. 97]). x= (x1, x2, . . . , xn)∈Rⁿ+logarithm majorizes G¯(x) = (G_n(x), G_n(x), . . . , G_n(x)).

Definition 2.2 ([7]). LetE ⊆Rⁿ+, thenE is said to be a logarithm convex set, if for anyx, y ∈E,α, β >0, α+β = 1, it havex^αy^β ∈E.

Remark 2. LetE ⊆Rⁿ+,lnE ={lnx|x∈E}. Thenx, y ∈E if only iflnx,lny ∈ lnE, andx^αy^β ∈Eif only ifαlnx+βlny∈ lnE, soEis a logarithm convex set if and only iflnE is a convex set.

Definition 2.3 ([10, p. 107]). LetE ⊆Rⁿ+, f :E → [0,+∞). Thenf is called an S−geometrically convex function, if for anyx, y ∈E ⊆Rⁿ+,lnxlny, we have

(2.1) f(x)≥f(y).

Andf is called anS−geometrically concave function, if the inequality (2.1) is re- versed.

Lemma 2.3 ([10, p. 108]). LetE ⊆ Rⁿ+ be a symmetric logarithm convex set with a nonempty interior,f :E → [0,+∞)be symmetric continuously differentiable on the interior ofE and continuous onE. Thenf is a S-geometrically convex function, if the following inequality

(2.2) (lnx₁ −lnx₂)

x₁ ∂f

∂x₁ −x₂ ∂f

∂x₂

≥0

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holds for allxin the interior ofE. Andf a is S-geometrically concave function, if the inequality (2.2) is reversed.

Proof. Let lnE = {lnx|x∈E}, then lnE is a symmetric convex set and has a nonempty interior. Again, letε > 0, g(x) = f(x) +ε with x ∈ E, and h(y) = lng(e^y) with y ∈ lnE = {lnx|x∈E}, then g : E → (0,+∞), h : lnE → (−∞,+∞). Further letx=e^y,

∂h

∂y₁ = ∂(lng(e^y))

∂y₁

= 1

g(e^y)· ∂(g(e^y))

∂y₁

= 1

g(x) ·∂(g(x))

∂x₁ ·e^y¹

= x₁ g(x) · ∂g

∂x₁. Similarly,

∂h

∂y₂ = x₂ g(x) · ∂g

∂x₂. According to inequality2.2,

(y1−y2) ∂h

∂y₁ − ∂h

∂y₂

= (lnx₁−lnx₂) g(x)

x1

∂g

∂x₁ −x2

∂g

∂x₂

= (lnx₁−lnx₂) g(x)

x1

∂f

∂x₁ −x2

∂f

∂x₂

≥0.

Then by Lemma2.1, we know thathis a S-convex function. For anyu, v ∈E with

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lnulnv, we have

h(lnu)≥h(lnv), lng e^ln^u

≥lng e^ln^v , and

g(u)≥g(v), f(u)≥f(v). Sof is a S- geometrically convex function.

If the inequality (2.2) is reversed, we similarly have thatf is a S-geometrically concave function.

The proof of Lemma2.3is completed.

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3. The Proof of Theorem 1.2

Proof of Theorem1.2. Ifn = 2, thenk = 2.

f(x) = P_n(x, k−1)−P_n(x, k) = x₁+x₂

2 −√

x₁x₂,

∂f

∂x₁ = 1 2 −1

2 rx₂

x₁, x1

∂f

∂x₁ = 1

2x1− 1 2

√x1x2,

(lnx₁−lnx₂)

x₁ ∂f

∂x₁ −x₂ ∂f

∂x₂

= (lnx₁−lnx₂)

x₁−x₂ 2

≥0.

According to Lemma2.3, ifn= 2, Theorem1.2is true.

Ifn≥3,k≥3, Lettingx¯= (x₃, x₄, . . . , x_n),En−2(¯x,0) = 1, we have f(x) = P_n(x, k−1)−P_n(x, k)

=

n k−1

−1

E_n(x, k−1)

!_k−1¹

− n

k −1

E_n(x, k) ¹_k

,

∂f

∂x1

= 1

k−1· n

k−1 ⁻_k−1¹

·(E_n(x, k−1))^k−1¹ ⁻¹ X

2≤i1<···<ij≤n k−2

Y

j=1

x_i_j

− 1 k ·n

k ⁻_k¹

·(E_n(x, k))¹^k⁻¹ X

2≤i₁<···<i_j≤n k−1

Y

j=1

x_i_j,

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x₁ ∂f

∂x₁ = 1 k−1·

n k−1

−_k−1¹

·(E_n(x, k−1))^k−1¹ ⁻¹[x₁En−2(¯x, k−2) +x₁x₂En−2(¯x, k−3)]

− 1 k ·n

k −¹

k ·(En(x, k))¹^k⁻¹[x1En−2(¯x, k−1) +x1x2En−2(¯x, k−2)], and

x₂ ∂f

∂x2

= 1

k−1· n

k−1 −_k−1¹

·(E_n(x, k−1))^k−1¹ ⁻¹[x₂En−2(¯x, k−2) +x₁x₂En−2(¯x, k−3)]

− 1 k ·n

k −¹

k ·(En(x, k))¹^k⁻¹[x2En−2(¯x, k−1) +x1x2En−2(¯x, k−2)]. So

(3.1) (lnx₁−lnx₂)

x₁ ∂f

∂x1

−x₂ ∂f

∂x2

= (lnx₁−lnx₂)· 1 k−1·

n k−1

⁻_k−1¹

·(E_n(x, k−1))^k−1¹ ⁻¹·(x₁−x₂)·En−2(¯x, k−2)

−(lnx₁−lnx₂)· 1 k ·n

k ⁻¹_k

·(E_n(x, k))¹^k⁻¹·(x₁−x₂)·En−2(¯x, k−1). On the other hand, by (1.2), we deduce

(x₁+x₂)En−2(¯x, k−1)·En−2(¯x, k−2) +x₁x₂

kE_n−2² (¯x, k−2)−(k−1)E_n−2(¯x, k−3)·E_n−2(¯x, k−1)

≥0,

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so

k·[(x₁+x₂)En−2(¯x, k−1) +x₁x₂En−2(¯x, k−2)]·En−2(¯x, k−2)

−(k−1)·[(x₁+x₂)E_n−2(¯x, k−2) +x₁x₂E_n−2(¯x, k−3)]·E_n−2(¯x, k−1)≥0, k·En(x, k)·En−2(¯x, k−2)−(k−1)·En(x, k−1)·En−2(¯x, k−1)≥0.

Again, according to (1.1), the following inequality holds.

k·P_n(x, k−1)·E_n(x, k)·En−2(¯x, k−2)

−(k−1)·P_n(x, k)·E_n(x, k−1)·En−2(¯x, k−1)≥0, then

1 k−1 ·

n k−1

−_k−1¹

·(E_n(x, k−1))^k−1¹ ⁻¹·En−2(¯x, k−2)

− 1 k ·n

k −¹_k

·(E_n(x, k))¹^k⁻¹·E_n−2(¯x, k−1)≥0.

Finally by (3.1), we can state that (lnx₁−lnx₂)

x₁ ∂f

∂x₁ −x₂ ∂f

∂x₂

≥0, Thus Theorem1.2holds by Lemma2.3.

Remark 3. If n ≥ 3, k = 2, we know thatf is neither a S-geometrically convex function nor a S-geometrically concave function.

Remark 4. Ifn = 2,orn ≥ 3,2 ≤ k −1 < k ≤ n, andxlogarithm majorizesy, according to Definition2.3, we can state that

Pn(x, k−1)−Pn(x, k)≥Pn(y, k−1)−Pn(y, k).

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By Lemma2.2and

P_n G¯(x), k−1

−P_n G¯(x), k

= 0, we know that Theorem1.2generalizes the Maclaurin-Inequality.

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