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volume 5, issue 3, article 69, 2004.

Received 03 March, 2004;

accepted 30 March, 2004.

Communicated by:D. Stefanescu

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Journal of Inequalities in Pure and Applied Mathematics

SOME INEQUALITIES FOR A CLASS OF GENERALIZED MEANS

KAI-ZHONG GUAN

Department of Mathematics and Physics Nanhua University, Hengyang, Hunan 421001 The Peoples’ Republic of China

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 054-04

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Some Inequalities for a Class of Generalized Means

Kai-Zhong Guan

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J. Ineq. Pure and Appl. Math. 5(3) Art. 69, 2004

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.

2000 Mathematics Subject Classification:05E05, 26D20.

Key words: Symmetric function ; Ky Fan inequality ; Harmonic-geometric mean in- equality.

1. Introduction

Let x = (x₁, x₂, . . . , x_n) be ann-tuple of positive numbers. The un-weighted arithmetic, geometric and harmonic means of x, denoted by A_n(x) , G_n(x), Hn(x), respectively, are defined as follows

A_n(x) = 1 n

n

X

i=1

x_i, G_n(x) =

n

Y

i=1

x_i

!_n¹

, H_n(x) = n Pn

i=1 1 xi

.

Assume that0≤x_i <1,1≤i≤nand define1−x= (1−x₁,1−x₂, . . . ,1− x_n).Throughout the sequel the symbolsA_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.

This result, commonly referred to as the Ky Fan inequality, has stimulated the interest of many researchers. New proofs, improvements and generaliza- tions 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 inequalityHn ≤Gn ≤An.Their result reads as follows.

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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, . . . , xn) =

"

Y

1≤i1<···ir≤n

r Pr

i=1x⁻¹_i_j

!#₍n¹ r)

.

ClearlyH_nⁿ(x) =H_n(x),H_n¹(x) = G_n(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;f is 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 Section3. In Section4, we investigate the Schur- convexity of the symmetric function. Several “Ky Fan” type inequalities are

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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−xi

Pn i=1

1 xi

≤

" Qn i=

1 1−xi

Qn i=

1 xi

#¹_n

or H_n(x)

H_n(1−x) ≤ G_n(x) G_n(1−x). Lemma 2.2. If 0 < x_i ≤ ¹₂,for alli = 1,2, . . . , n+ 1,andS_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

Sn+1− _x¹

i





n

≤



 Qn+1

i=1

S_n+1− _1−x¹

i

Qn+1

i=1

Sn+1− _x¹

i





1 n+1

.

Proof. Inequality (2.2) is equivalent to the following

nlnS_n+1

S_n+1 ≤ 1 n+ 1 ln



 Qn+1

i=1

S_n+1−_1−x¹

i

Qn+1

i=1(S_n+1− _x¹

i)



 Since0< x_i ≤ ¹₂, and1−x_i ≥x_i, 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−xj+1· · ·_1−x¹

n+1

1 x1 · · ·_x¹

j−1

1

xj+1· · ·_x¹

n+1

.

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By the above inequality and Lemma2.1, we have 1

n+ 1 ln

n+1

Y

i=1

S_n+1−_1−x¹

i

S_n+1− _x¹

i

≥ 1 n+ 1ln

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+ 1 ln



 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, . . . , xn) be symmetric and have continuous partial derivatives on Iⁿ, where I is an open interval. Then f :Iⁿ→Ris Schur-convex if and only if

(2.3) (x_i −x_j)

∂f

∂x_i − ∂f

∂x_j

≥0

on Iⁿ. It is strictly Schur-convex if (2.3) is a strict inequality for x_i 6= x_j, 1≤i, j ≤n.

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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 of f(x) does not have to be a Cartesian productIⁿ.Lemma2.3remains 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 implies P x ∈ Afor any n×n permutation matrixP.

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3. Refinement of the Harmonic-Geometric Mean Inequality

The goal of this section is to obtain the basic inequality of H_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<···<ir+1≤n

ln

"

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

k=1x⁻¹_i

k −Pr+1

j=1x⁻¹_i_j

#

= X

1≤i1<···<ir+1≤n

ln







r hPr+1

j=1

Pr+1 k=1x⁻¹_i

k −x⁻¹_i

j

i.

(r+ 1)







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≤ X

1≤i₁<···<i_r+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_i_j

#_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_i

j

#

= 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_i

k

= (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),

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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 3.1. The corollary refines the harmonic-geometric mean inequality.

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4. Schur-convexity of the Function H

_n^r

(x)

In this section, we investigate the Schur-convexity of the function H_n^r(x), and establish several 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 functionH_n^r(x)is Schur-concave inRⁿ+.

Proof. It is clear thatH_n^r(x)is symmetric and has continuous partial derivatives onRⁿ+. By Lemma2.3, 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<···<ir≤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_i

j





+ X

3≤i1<···<ir−2≤n

1

x⁻¹₁ +x⁻¹₂ +Pr−2 j=1x⁻¹_i

j



.

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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_i_j





+ 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_i

j



· 1 x²₂

+ X

3≤i₁<···<ir−2≤n

1

x⁻¹₁ +x⁻¹₂ +Pr−2 j=1x⁻¹_i

1 x²₁ − 1

x²₂





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=−(x₁−x₂)²





(x1+x2) x²₁x²₂

X

3≤i1<···<ir−2≤n

1 x⁻¹₁ +x⁻¹₂ +Pr−2

j=1x⁻¹_i

j

+ 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. Let x_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₍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 Theorem4.1, we obtain H_n^r

c+x nc+s

≥H_n^rx s

.

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Or

H_n^r(c+x)

H_n^r(x) ≥nc

s + 1₍¹n r)

.

Corollary 4.3. Let x_i > 0, i = 1,2, . . . , n, n ≥ 2, and Pn

i=1x_i = s, c ≥ s, then

(4.2) H_n^r(c−x)

H_n^r(x) ≥nc

s −1₍¹n 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 Theorem4.1, 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

¹

(ⁿ_r)

.

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Remark 4.1. 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)

Hn(x) ≥(n−1), G_n(1−x) Gn(x) ≥ √ⁿ

n−1.

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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

ϕ_r = 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 Lemma2.2and the monotonicity of the functiony= lnx, we have n

r+ 1

lnφ_r+1 = X

1≤i₁<···<i_r+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

ln





r+1

Y Pr+1

k=1 1

1−x_ik −_1−x¹

ij

Pr+1 1 − ¹





1 r(r+1)

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= 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 Theorem3.1, we can derive n

r+ 1

lnφr+1 ≤ 1

r(r+ 1)(n−r) n

r

lnφr = 1 r

n r+ 1

lnφr. Thus

(φ_r)¹^r ≥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 5.1. By Theorem5.1, 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 Lemma2.1and the proof of Theorem3.1, we have the following Theorem 5.2. If0< x_i ≤ ¹₂, i = 1,2, . . . , n,then

(5.3)

Qn i=1(x_i) Qn

i=1(1−x_i) ≤ H_n(x)

H_n(1−x) ≤ G_n(x)

G_n(1−x) ≤ A_n(x) A_n(1−x).

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The inequality (5.3) generalizes the inequality (1.2).

Theorem 5.3. If0< xi ≤ ¹₂, i = 1,2, . . . , n,then H_n^r(x)

H_n^r(1−x) ≤ H_n¹(x) H_n¹(1−x) (5.4)

= 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 Lemma2.1and the monotonicity of the functiony= lnx, we have n

r

lnφ_r = 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

.

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By knowledge of combinations, we can easily find n

r

lnφ_r≤ 1 r ln

" _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−x_i

1 xi

#

= 1 r

n−1 r−1

lnφ₁ =n r

lnφ₁. Thus

φr ≤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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References

[1] E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer-Verlag, New York, 1965.

[2] H. ALZER, The inequality of Ky Fan and related results, Acta. Appl. Math., 38 (1995), 305–354.

[3] J. SÁNDOR, On an inequaliy of Ky Fan, Babe¸s-Bolyai Univ., Fac. Math.

Phys., Res. Semin., 7 (1990), 29–34.

[4] J. SÁNDOR AND T. TRIF, A new refinement of the Ky Fan inequality, Math. Inequal. Appl., 2 (1999), 529–533.

[5] W.-L. WANGANDP.-F. WANG, A class of inequalities for symmetric func- tions (in chinese), Acta. Math .Sinica, 27 (1984), 485–497.

[6] A.W. ROBERTSANDD.E. VARBERG, Convex Function, Academic Press, New York, San Francisco, London, 1973.

[7] A.W. MARSHALL AND I. OLKIN, Inequalities: Theory of Majorization and Its Applications, Academic Press, 1979.

[8] D.S. MITRINOVI ´C, Analytic Inequalities, Springer-Verlag, New York, 1970.

[9] HUANNAN SHI, Refinement and generalization of a class of inequalities for symmetric functions (in Chinese), Mathematics in Practices and The- ory, 4 (1999), 81–84.