Ky Fan inequality

http://jipam.vu.edu.au/

Volume 5, Issue 3, Article 69, 2004

SOME INEQUALITIES FOR A CLASS OF GENERALIZED MEANS

KAI-ZHONG GUAN

DEPARTMENT OFMATHEMATICS ANDPHYSICS

NANHUAUNIVERSITY, HENGYANG, HUNAN421001 THEPEOPLES’ REPUBLIC OFCHINA

kaizhongguan@yahoo.com.cn

Received 15 March, 2004; accepted 04 April, 2004 Communicated by D. Stefanescu

ABSTRACT. In this paper, we define a symmetric function, show its properties, and establish several analytic inequalities, some of which are "Ky Fan" type inequalities. The harmonic- geometric mean inequality is refined.

Key words and phrases: Symmetric function ; Ky Fan inequality ; Harmonic-geometric mean inequality.

2000 Mathematics Subject Classification. 05E05, 26D20.

1. INTRODUCTION

Let x = (x₁, x₂, . . . , x_n) be an n-tuple of positive numbers. The un-weighted arithmetic, geometric and harmonic means ofx, denoted byA_n(x),G_n(x),H_n(x), respectively, are defined as follows

An(x) = 1 n

n

X

i=1

xi, Gn(x) =

n

Y

i=1

xi

!¹_n

, Hn(x) = n Pn

i=1 1 xi

.

Assume that 0 ≤ x_i < 1, 1 ≤ i ≤ n and define 1− x = (1− x₁,1− x₂, . . . ,1 −x_n).

Throughout the sequel the symbols A_n(1−x), G_n(1−x), H_n(1−x) will stand for the un- weighted arithmetic, geometric, harmonic means of1−x.

A remarkable new counterpart of the inequalityG_n ≤A_nhas been published in [1].

Theorem 1.1. If0< x_i ≤ ¹₂, for alli= 1,2, . . . , n,then

(1.1) G_n(x)

Gn(1−x) ≤ A_n(x) An(1−x) with equality if and only if all thex_i are equal.

ISSN (electronic): 1443-5756

c 2004 Victoria University. All rights reserved.

A Project Supported by Scientific Research Fund of Hunan Provincial Education Department (China)(granted 03C427).

054-04

This result, commonly referred to as the Ky Fan inequality, has stimulated the interest of many researchers. New proofs, improvements and generalizations of the inequality (1.1) have been found. For more details, interested readers can see [2], [3] and [4].

W.-L. Wang and P.-F. Wang [5] have established a counterpart of the classical inequality H_n≤G_n≤A_n.Their result reads as follows.

Theorem 1.2. If0< x_i ≤ ¹₂, for alli= 1,2, . . . , n,then

(1.2) H_n(x)

H_n(1−x) ≤ G_n(x)

G_n(1−x) ≤ A_n(x) A_n(1−x).

All kinds of means about numbers and their inequalities have stimulated the interest of many researchers. Here we define a new mean, that is:

Definition 1.1. Letx∈ Rⁿ+ = {x|x= (x₁, x₂, . . . , x_n)|x_i >0, i= 1,2, . . . , n}, we define the symmetric function as follows

H_n^r(x) =H_n^r(x₁, x₂, . . . , x_n) =

"

Y

1≤i1<···ir≤n

r Pr

i=1x⁻¹_ij

!# ¹

(ⁿ_r)

.

ClearlyH_nⁿ(x) = Hn(x),H_n¹(x) =Gn(x), where ⁿ_r

= _r!(n−r)!^n! .

The Schur-convex function was introduced by I. Schur in 1923 [7]. Its definition is as follows:

Definition 1.2. f :Iⁿ →R(n >1)is called Schur-convex ifx≺y, then

(1.3) f(x)≤f(y)

for allx, y ∈Iⁿ =I×I×· · ·×I(n copies).It is called strictly Schur-convex if the inequality is strict;fis called Schur-concave (resp. strictly Schur-concave) if the inequality (1.3) is reversed.

For more details, interested readers can see [6], [7] and [8].

The paper is organized as follows. A refinement of harmonic-geometric mean inequality is obtained in Section 3. In Section 4, we investigate the Schur-convexity of the symmetric function. Several “Ky Fan” type inequalities are obtained in Section 5.

2. LEMMAS

In this section, we give the following lemmas for the proofs of our main results.

Lemma 2.1. ([5]) If0< x_i ≤ ¹₂, for alli= 1,2, . . . , n,then

(2.1)

Pn i=1

1 1−x_i

Pn i=1

1 xi

≤

" Qn i=

1 1−x_i

Qn i=

1 xi

#_n¹

or H_n(x)

H_n(1−x) ≤ G_n(x) G_n(1−x). Lemma 2.2. If 0 < x_i ≤ ¹₂, for all i = 1,2, . . . , n+ 1, and S_n+1 = Pn+1

i=1 1

xi, S_n+1 = Pn+1

i=1 1

1−xi, then

(2.2)



 Pn+1

i=1

S_n+1− _1−x¹

i

Pn+1

i=1

S_n+1− _x¹

i





n

≤



 Qn+1

i=1

S_n+1− _1−x¹

i

Qn+1

i=1

S_n+1− _x¹

i





1 n+1

.

Proof. Inequality (2.2) is equivalent to the following

nlnSn+1

S_n+1 ≤ 1 n+ 1ln



 Qn+1

i=1

S_n+1− _1−x¹

i

Qn+1

i=1(S_n+1− _x¹

i)



 Since0< xi ≤ ¹₂, and1−xi ≥xi, it follows that

S_n+1− _1−x¹

j

S_n+1− _x¹

j

=

1

1−x1 +· · ·+ _1−x¹

j−1 +_1−x¹

j+1 +· · ·+_1−x¹

n+1

1

x1 +· · ·+_x¹

j−1 +_x¹

j+1 +· · ·+_x¹

n+1

≥

1

1−x₁ · · ·_1−x¹

j−1

1

1−x_j+1 · · ·_1−x¹

n+1

1

x1 · · ·_x¹

j−1

1

xj+1 · · ·_x¹

n+1

.

By the above inequality and Lemma 2.1, we have 1

n+ 1ln

n+1

Y

i=1

S_n+1− _1−x¹

i

S_n+1− _x¹

i

≥ 1 n+ 1 ln

n+1

Y

i=1

1 1−x_i

1 x_i

n

=nln

n+1

Y

i=1

1 1−x_i

1 x_i

n+1

≥nln

1

1−x₁ +· · ·+_1−x¹

n+1

1

x1 +· · ·+_x¹

n+1

,

or

nlnS_n+1

S_n+1 ≤ 1 n+ 1ln



 Qn+1

i=1

S_n+1− _1−x¹

i

Qn+1

i=1

S_n+1−_x¹

i



.

Lemma 2.3. [6, p. 259]. Let f(x) = f(x₁, x₂, . . . , x_n) be symmetric and have continuous partial derivatives onIⁿ,whereI is an open interval. Thenf :Iⁿ→Ris Schur-convex if and only if

(2.3) (x_i−x_j)

∂f

∂x_i − ∂f

∂x_j

≥0

onIⁿ. It is strictly Schur-convex if (2.3) is a strict inequality forx_i 6=x_j,1≤i, j ≤n.

Sincef(x)is symmetric, Schur’s condition can be reduced as [7, p. 57]

(2.4) (x₁−x₂)

∂f

∂x₁ − ∂f

∂x₂

≥0,

and f is strictly Schur-convex if (2.4) is a strict inequality for x₁ 6= x₂.The Schur condition that guarantees a symmetric function being Schur-concave is the same as (2.3) or (2.4) except the direction of the inequality.

In Schur’s condition, the domain off(x)does not have to be a Cartesian productIⁿ.Lemma 2.3 remains true if we replaceIⁿby a setA⊆Rⁿwith the following properties ([7, p. 57]):

(i) Ais convex and has a nonempty interior;

(ii) A is symmetric in the sense that x ∈ A impliesP x ∈ A for any n×n permutation matrixP.

3. REFINEMENT OF THEHARMONIC-GEOMETRICMEANINEQUALITY

The goal of this section is to obtain the basic inequality ofH_n^r(x), and give a refinement of the Harmonic-Geometric mean inequality.

Theorem 3.1. Letx∈Rⁿ+={x|x= (x₁, x₂, . . . , x_n)|x_i >0, i= 1,2, . . . , n},then (3.1) H_n^r+1(x)≤H_n^r(x), r = 1,2, . . . , n−1.

Proof. By the arithmetic-geometric mean inequality and the monotonicity of the functiony = lnx, we have

n r+ 1

lnH_n^r+1(x) = X

1≤i1<···<ir+1≤n

ln r+ 1 Pr+1

j=1x⁻¹_i

j

= X

1≤i₁<···<i_r+1≤n

ln

"

(r+ 1)r (r+ 1)Pr+1

k=1x⁻¹_i

k −Pr+1

j=1x⁻¹_ij

#

= X

1≤i₁<···<i_r+1≤n

ln







r hPr+1

j=1

Pr+1 k=1x⁻¹_i

k −x⁻¹_ij i.

(r+ 1)







≤ X

1≤i1<···<ir+1≤n

ln







r Qr+1

j=1

Pr+1 k=1x⁻¹_i

k −x⁻¹_i

j

_r+1¹







= X

1≤i₁<···<i_r+1≤n

ln

"_r+1 Y

j=1

r Pr+1

k=1x⁻¹_i

k −x⁻¹_ij

#_r+1¹

= 1

r+ 1

X

1≤i1<···<ir+1≤n

"_r+1 X

j=1

ln r

Pr+1 k=1x⁻¹_i

k −x⁻¹_ij

#

= 1

r+ 1

n

X

j=1

i1,...,ir6=j

X

1≤i₁<···<ir≤n

ln r Pr

k=1x⁻¹_i

k

.

Let

S_j =

i1,...,ir6=j

X

1≤i1<···<ir≤n

ln r Pr

k=1x⁻¹_i

k

, j = 1,2, . . . , n.

We can easily get

n

X

j=1

S_j = (n−r) X

1≤i1<···<ir≤n

ln r Pr

k=1x⁻¹_ik = (n−r)n r

lnH_n^r(x).

Thus

n r+ 1

lnH_n^r+1(x)≤ n−r r+ 1

n r

lnH_n^r(x) = n

r+ 1

lnH_n^r(x), or

H_n^r+1(x)≤H_n^r(x), r = 1,2, . . . , n−1.

Corollary 3.2. Letx∈Rⁿ+ ={x|x= (x1, x2, . . . , xn)|xi >0, i= 1,2, . . . , n}, then

(3.2) H_n(x)≤H_nⁿ⁻¹(x)≤ · · · ≤H_n²(x)≤H_n¹(x) =G_n(x).

Remark 3.3. The corollary refines the harmonic-geometric mean inequality.

4. SCHUR-CONVEXITY OF THE FUNCTIONH_n^r(x)

In this section, we investigate the Schur-convexity of the functionH_n^r(x), and establish sev- eral analytic inequalities by use of the theory of majorization.

Theorem 4.1. LetRⁿ+ = {x|x = (x₁, x₂, . . . , x_n)|x_i > 0, i = 1,2, . . . , n}, then the function H_n^r(x)is Schur-concave inRⁿ+.

Proof. It is clear that H_n^r(x) is symmetric and has continuous partial derivatives on Rⁿ+. By Lemma 2.3, we only need to prove

(x1−x2)

∂H_n^r(x)

∂x₁ −∂H_n^r(x)

∂x₂

≤0.

As matter of fact, we can easily derive lnH_n^r(x) = 1

n r

X

2≤i₁<···<i_r≤n

ln r Pr

j=1x⁻¹_i

j

+ X

2≤i₁<···<ir−1≤n

ln r

x⁻¹₁ +Pr−1 j=1x⁻¹_i

j

.

DifferentiatinglnH_n^r(x)with respect tox₁, we have

∂H_n^r(x)

∂x₁ = H_n^r(x)

n r





X

2≤i₁<···<ir−1≤n

1 x⁻¹₁ +Pr−1

j=1x⁻¹_i

j



· 1 x²₁

= H_n^r(x)

n r

· 1 x²₁









X

3≤i₁<···<ir−1≤n

1 x⁻¹₁ +Pr−1

j=1x⁻¹_ij





+ X

3≤i1<···<ir−2≤n

1

x⁻¹₁ +x⁻¹₂ +Pr−2 j=1x⁻¹_i

j



.

Similar to the above, we can also obtain

∂H_n^r(x)

∂x₂ = H_n^r(x)

n r





X

2≤i₁<···<ir−1≤n

1 x⁻¹₂ +Pr−1

j=1x⁻¹_i

j



· 1 x²₂

= H_n^r(x)

n r

· 1 x²₂









X

3≤i1<···<ir−1≤n

1 x⁻¹₂ +Pr−1

j=1x⁻¹_ij





+ X

3≤i1<···<ir−2≤n

1

x⁻¹₁ +x⁻¹₂ +Pr−2 j=1x⁻¹_i

j



.

Thus

(x₁ −x₂)

∂H_n^r(x)

∂x₁ − ∂H_n^r(x)

∂x₂

= (x₁−x₂)H_n^r(x)

n r









X

2≤i₁<···<ir−1≤n

1 x⁻¹₁ +Pr−1

j=1x⁻¹_i

j



· 1 x²₁

−





X

2≤i1<···<ir−1≤n

1 x⁻¹₂ +Pr−1

j=1x⁻¹_ij



· 1 x²₂

+ X

3≤i1<···<ir−2≤n

1

x⁻¹₁ +x⁻¹₂ +Pr−2 j=1x⁻¹_i

j

1

x²₁ − 1 x²₂





=−(x₁−x₂)²





(x₁+x₂) x²₁x²₂

X

3≤i1<···<ir−2≤n

1 x⁻¹₁ +x⁻¹₂ +Pr−2

j=1x⁻¹_ij

+ X

2≤i1<···<ir−1≤n

1 + (x₁+x₂)Pr−1 j=1x⁻¹_i

j

x²₁x²₂

x⁻¹₁ +Pr−1 j=1x⁻¹_i

j x⁻¹₂ +Pr−1 j=1x⁻¹_i

j





≤0.

Corollary 4.2. Letx_i >0, i= 1,2, . . . , n,n ≥2,andPn

i=1x_i =s,c >0, then

(4.1) H_n^r(c+x)

H_n^r(x) ≥nc s + 1

¹

(ⁿ_r)

, r = 1,2, . . . , n,

wherec+x= (c+x₁, c+x₂, . . . , c+x_n).

Proof. By [9], we have c+x nc+s =

c+x₁

nc+s, . . . , c+x_n nc+s

≺x₁

s , . . . ,x_n s

= x s.

Using Theorem 4.1, we obtain H_n^r

c+x nc+s

≥H_n^rx s

. Or

H_n^r(c+x)

H_n^r(x) ≥nc

s + 1₍¹n r)

.

Corollary 4.3. Letx_i >0, i= 1,2, . . . , n,n≥2, andPn

i=1x_i =s,c≥s, then

(4.2) H_n^r(c−x)

H_n^r(x) ≥nc

s −1₍n¹ r)

, r = 1,2, . . . , n, wherec−x= (c−x₁, c−x₂, . . . , c−x_n).

Proof. By [9], we have c−x nc−s =

c−x₁

nc−s, . . . ,c−x_n nc−s

≺x₁

s , . . . ,x_n s

= x s. Using Theorem 4.1, we obtain

H_n^r

c−x nc−s

≥H_n^rx s

, or

H_n^r(c−x)

H_n^r(x) ≥nc

s −1₍n¹ r)

.

Remark 4.4. Letc=s= 1, we can obtain

H_n^r(1−x)

H_n^r(x) ≥(n−1)

1

(ⁿ_r), r = 1,2, . . . , n.

In particular,

H_n(1−x)

H_n(x) ≥(n−1), G_n(1−x) G_n(x) ≥ √ⁿ

n−1.

5. SOME“KY FAN” TYPE INEQUALITIES

In this section, some “Ky Fan” type inequalities are established, the Ky Fan inequality is generalized.

Theorem 5.1. Assume that0< x_i ≤ ¹₂, i = 1,2, . . . , n,then

(5.1) H_n^r+1(x)

H_n^r+1(1−x) ≤

H_n^r(x) H_n^r(1−x)

¹_r

, r = 1,2, . . . , n−1.

Proof. Set

ϕr = H_n^r(x)

H_n^r(1−x) = Y

1≤i1<···<ir≤n



 Pr

j=1 1 1−x_ij

Pr j=1

1 x_ij





1

(ⁿ_r)

.

By Lemma 2.2 and the monotonicity of the functiony= lnx, we have n

r+ 1

lnφ_r+1 = X

1≤i1<···<ir+1≤n

ln Pr+1

j=1 1 1−x_ij

Pr+1 j=1

1 x_ij

= X

1≤i1<···<ir+1≤n

ln Pr+1

j=1

Pr+1 k=1

1

1−x_ik − _1−x¹

ij

Pr+1

j=1

Pr+1 k=1

1 x_ik − _x¹

ij

≤ X

1≤i₁<···<i_r+1≤n

ln





r+1

Y

j=1

Pr+1 k=1

1

1−x_ik − _1−x¹

ij

Pr+1 k=1

1 x_ik − _x¹

ij





1 r(r+1)

= 1

r(r+ 1)

n

X

j=1

i1,...,ir6=j

X

1≤i₁<···<ir≤n

ln Pr

k=1 1 1−x_ik

Pr k=1

1 x_ik

.

Similar to Theorem 3.1, we can derive n

r+ 1

lnφ_r+1 ≤ 1

r(r+ 1)(n−r)n r

lnφ_r = 1 r

n r+ 1

lnφ_r.

Thus

(φ_r)^1r ≥φ_r+1, or

H_n^r+1(x) H_n^r+1(1−x) ≤

H_n^r(x) H_n^r(1−x)

¹_r

, r = 1,2, . . . , n−1.

Remark 5.2. By Theorem 5.1, we can obtain

(5.2) H_n²(x)

H_n²(1−x) ≤ H_n¹(x)

H_n¹(1−x) = G_n(x)

G_n(1−x) ≤ A_n(x) A_n(1−x). This is a generalization of the “Ky Fan” inequality.

By Lemma 2.1 and the proof of Theorem 3.1, we have the following Theorem 5.3. If0< x_i ≤ ¹₂, i= 1,2, . . . , n,then

(5.3)

Qn i=1(x_i) Qn

i=1(1−xi) ≤ H_n(x)

Hn(1−x) ≤ G_n(x)

Gn(1−x) ≤ A_n(x) An(1−x). The inequality (5.3) generalizes the inequality (1.2).

Theorem 5.4. If0< x_i ≤ ¹₂, i= 1,2, . . . , n,then

(5.4) H_n^r(x)

H_n^r(1−x) ≤ H_n¹(x)

H_n¹(1−x) = G_n(x)

G_n(1−x) ≤ A_n(x)

A_n(1−x), r= 2, . . . , n.

Proof. Set

ϕ_r = H_n^r(x)

H_n^r(1−x) = Y

1≤i₁<···<i_r≤n



 Pr

j=1 1 1−x_ij

Pr j=1

1 x_ij





1

(ⁿ_r)

.

By Lemma 2.1 and the monotonicity of the functiony= lnx, we have n

r

lnφr = X

1≤i1<···<ir+1≤n

ln Pr

j=1 1 1−x_ij

Pr j=1

1 x_ij

≤ X

1≤i1<···<ir+1≤n

ln



 Qr

j=1 1 1−x_ij

Qr j=1

1 x_ij





1 r

= 1 r

X

1≤i1<···<ir≤n r

X

j=1

ln

1 1−x_ij

1 x_ij

.

By knowledge of combination, we can easily find n

r

lnφr ≤ 1 rln

" _n Y

i=1 1 1−x_i

1 xi

#(ⁿ⁻¹r−1)

= 1 r

n−1 r−1

ln

" _n Y

i=1 1 1−xi

1 xi

#

= 1 r

n−1 r−1

lnφ₁ =n r

lnφ₁. Thus

φ_r ≤φ₁, r= 2, . . . , n, or

(5.5) H_n^r(x)

H_n^r(1−x) ≤ G_n(x)

G_n(1−x), r= 2, . . . , n.

The inequality (5.5) generalizes the “Ky Fan” inequality.

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