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volume 4, issue 2, article 39, 2003.

Received 21 October, 2002;

accepted 28 March, 2003.

Communicated by:F. Qi

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Journal of Inequalities in Pure and Applied Mathematics

THE INEQUALITIES G≤L≤I ≤A INn VARIABLES

ZHEN-GANG XIAO AND ZHI-HUA ZHANG

Department of Mathematics,

Hunan Institute of Science and Technology, Yueyang City, Hunan 414006,

CHINA.

E-Mail:xzgzzh@163.com

Zixing Educational Research Section, Chenzhou City,

Hunan 423400, CHINA.

EMail:zxzh1234@163.com

c

2000Victoria University ISSN (electronic): 1443-5756 110-02

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The InequalitiesG≤L≤I≤A innVariables

Zhen-Gang Xiao and Zhi-Hua Zhang

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Abstract

In the short note, the inequalitiesG≤L≤I≤Afor the geometric, logarithmic, identric, and arithmetic means innvariables are proved.

2000 Mathematics Subject Classification:26D15.

Key words: Inequality, Arithmetic mean, Geometric mean, Logarithmic mean, Iden- tric mean,nvariables, Van der Monde determinant.

The authors would like to thank Professor Feng Qi and the anonymous referee for some valuable suggestions which have improved the final version of this paper.

1. Introduction

For positive numbersa_i,1≤i≤2, let A=A(a₁, a₂) = a1+a2

2 ; (1.1)

G=G(a₁, a₂) =√ a₁a₂; (1.2)

I =I(a₁, a₂) =





 exp

a2lna2 −a1lna1

a₂ −a₁ −1

, a₁ < a₂,

a₁, a₁ =a₂;

(1.3)

L=L(a₁, a₂) =







a2−a1

lna₂−lna₁, a₁ < a₂, a₁, a₁ =a₂. (1.4)

These are respectively called the arithmetic, geometric, identric, and logarithmic means.

The logarithmic mean [1,3], somewhat supprisingly, has applications to the economical index analysis. K.B. Stolarsky first introduced the identric meanI and provedG≤L≤I ≤Ain two variables in [4]. See [2] also.

The purpose of this short note is to prove the inequalitiesG ≤ L ≤ I ≤ A of the geometric, logarithmic, identric, and arithmetic means innvariables.

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2. Definitions and the Main Result

Let a = (a₁, a₂, . . . , a_n)and a_i > 0for1 ≤ i ≤ n, then the arithmetic, geometric, identric, and logarithmic means in nvariables are defined respectively as follows

A=A(a) =A(a₁, . . . , a_n) = a₁+a₂+· · ·+a_n

n ,

(2.1)

G=G(a) =G(a₁, . . . , a_n) = √ⁿ

a₁a₂· · ·a_n, (2.2)

I =I(a) =I(a₁, . . . , a_n) (2.3)

= exp

"

1 V(a)

n

X

i=1

(−1)ⁿ⁺ⁱaⁿ⁻¹_i V_i(a) lna_i−m

# ,

L=L(a) =L(a₁, . . . , a_n) = (n−1)!

V(lna)

n

X

i=1

(−1)ⁿ⁺ⁱa_iV_i(lna), (2.4)

wherelna = (lna₁, . . . ,lna_n),a_i 6=a_j fori6=j,

(2.5) V(a) =

1 1 . . . 1

a₁ a₂ . . . a_n a²₁ a²₂ . . . a²_n . . . . aⁿ⁻¹₁ aⁿ⁻¹₂ . . . aⁿ⁻¹_n

= Y

1≤j<i≤n

(a_i−a_j)

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is the determinant of Van der Monde’s matrix of then-th order,

(2.6) Vi(a) =

1 1 . . . 1 1 . . . 1

a₁ a₂ . . . ai−1 a_i+1 . . . a_n a²₁ a²₂ . . . a²_i−1 a²_i+1 . . . a²_n . . . . aⁿ⁻²₁ aⁿ⁻²₂ . . . aⁿ⁻²_i−1 aⁿ⁻²_i+1 . . . aⁿ⁻²_n

,

m =Pn−1 k=1 1

k, and1≤i≤n.

The main result of this short note can be stated as

Theorem 2.1. Let a = (a₁, a₂, . . . , a_n)and a_i > 0 for 1 ≤ i ≤ n, then the inequalities

(2.7) G(a)≤L(a)≤I(a)≤A(a)

of the geometric, logarithmic, identric, and arithmetic means in n variables hold. The equalities in (2.7) are valid if and only ifa₁ =a₂ =· · ·=a_n.

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3. Proof of Theorem 2.1

To prove inequalities in (2.7), we introduce the following means

Ir(a) = Y

i1+i2+···+in=n+r−1 i1,i2,...,in≥1

" _n X

k=1

i_k n+r−1ak

# ¹ (^n+r−2r−1 )

, (3.1)

I_r⁰(a) = Y

i1+i2+···+i_n=r i1,i2,...,in≥0

" _n X

k=1

i_k ra_k

# ¹ (^n+r−1r )

, (3.2)

Lr(a) = 1

n+r−1 r

X

i1+i2+···+in=r i1,i2,...,in≥0

n

Y

k=1

aⁱ_k^k^/r, (3.3)

L⁰_r(a) = 1

n+r−2 r−1

X

i1+i2+···+in=n+r−1 i1,i2,...,in≥1

n

Y

k=1

aⁱ_k^k^/(n+r−1), (3.4)

wherea= (a₁, a₂, . . . , a_n)anda_i >0for1≤i≤n.

Lemma 3.1. Leta = (a1, a2, . . . , an)andai >0for1≤i≤n, then we have 1. I₁(a) =L₁(a) = A(a);

2. I₁⁰(a) =L⁰₁(a) = G(a);

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3. For1≤j ≤n−1and

(3.5) π_j = Y

i1+i2+···+i_n=n+r−1 i_k₁=i_k₂=···=i_kj=0,for the resti_k≥1

n

X

k=1

i_k

n+r−1a_k,

we have

(3.6) I_n+r−1⁰ (a) =

n−1

Y

j=1

π¹

(^2n+r−2n+r−1)

j

I_r(a)(^n+r−2r−1 )

(^2n+r−2n+r−1)

;

4. For1≤j ≤n−1and

(3.7) δj = X

i1+i2+···+in=n+r−1 ik1=ik2=···=i_kj=0,for the restik≥1

n

Y

k=1

aⁱ_k^k^/(n+r−1),

we have

(3.8) L_n+r−1(a) = 1

2n+r−2 n+r−1

"_n−1 X

j=1

δ_j+

n+r−2 r−1

L⁰_r(a)

#

;

5. Ifr ∈N, then

(a) Ir(a)≥Ir+1(a), (b) I_r⁰(a)≤I_r+1⁰ (a),

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(c) L_r(a)≥L_r+1(a), (d) L⁰_r(a)≤L⁰_r+1(a), (e) I_r(a)≥L_r(a),

(f) I_r⁰(a)≥L⁰_r(a),

where equalities above hold if and only ifa₁ =a₂· · ·=a_n.

Proof. The formula (3.6) follows from standard arguments and formulas (3.1) and (3.2).

Ifr ∈Nandi₁+i₂+· · ·+i_n =n+r−1, then

n

X

k=1

i_k

n+r−1a_k =

n

X

k=1

i_k

n+r · n+r n+r−1a_k

= n+r n+r−1

n

X

k=1

ik

n+rak

= 1

n+r−1

" _n X

j=1

i_j+ 1

# _n X

k=1

i_k n+ra_k

= 1

n+r−1

" _n X

j=1

i_j

n

X

k=1

i_k n+ra_k+

n

X

k=1

i_k n+ra_k

#

= 1

n+r−1

" _n X

j=1

i_j

n

X

k=1

i_k n+ra_k+

n

X

j=1

i_j n+ra_j

#

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=

n

X

j=1

i_j n+r−1

" _n X

k=1

i_ka_k

n+r + a_j n+r

# .

By using the weighted arithmetic-geometric mean inequality, we have

n

X

k=1

i_k

n+r−1ak ≥

n

Y

j=1

" _n X

k=1

i_ka_k

n+r + a_j n+r

#ij/(n+r−1)

,

and then

Y

i1+i2+···+in=n+r−1 i1,i2,...,in≥1

n

X

k=1

i_k n+r−1a_k (3.9)

≥ Y

i1+i2+···+in=n+r−1 i1,i2,...,in≥1

n

Y

j=1

" _n X

k=1

i_ka_k

n+r + a_j n+r

#ij/(n+r−1)

=

n

Y

j=1

Y

i1+i2+···+in=n+r−1 i1,i2,...,in≥1

" _n X

k=1

i_ka_k

n+r + a_j n+r

#ij/(n+r−1)

=

n

Y

j=1

Y

ν1+ν2+···+νn=n+r ν1,ν2,...,νj−1,νj+1,...,νn≥1;νj≥2

" _n X

k=1

ν_k n+ra_k

#(νj−1)/(n+r−1)

= Y

ν1+ν2+···+νn=n+r ν1,ν2,...,νn≥1

" _n X

k=1

ν_k n+rak

#^Pⁿ_j=1(νj−1)/(n+r−1)

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= Y

ν1+ν2+···+νn=n+r ν1,ν2,...,νn≥1

" _n X

k=1

ν_k n+rak

#(^Pⁿj=1νj−n)/^(n+r−1)

= Y

ν1+ν2+···+νn=n+r ν1,ν2,...,νn≥1

" _n X

k=1

ν_k n+rak

#^r/(n+r−1)

= Y

ν1+ν2+···+νn=n+r ν1,ν2,...,νn≥1

" _n X

k=1

νk

n+rak

#(^n+r−2r−1 )/(^n+r−1_r ) ,

notice that the result from line 4 to line 5 in (3.9) follows from a simple fact that

" _n X

k=1

ν_k n+ra_k

#(νj−1)/(n+r−1)

= 1 for ν_j = 1.

The equalities above are valid if and only if

n

X

k=1

i_ka_k

n+r + a₁ n+r =

n

X

k=1

i_ka_k

n+r + a₂

n+r =· · ·=

n

X

k=1

i_ka_k

n+r + a_n n+r, which is equivalent toa₁ =a₂ =· · ·=a_n. This implies thatI_r(a)≥I_r+1(a).

The inequality I_r(a) ≥ L_r(a) follows easily from the generalized Hölder inequality

(3.10) Y

i +i +···+i =r+1

" _n X

j=1

i_ja_j

#¹_r

≥ X

i +i +···+i =r

" _n Y

j=1

aⁱ_j^j

#¹_r .

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The proofs of other formulas and inequalities will be left to the readers.

Lemma 3.2. Leta = (a₁, a₂, . . . , a_n)anda_i >0for1≤i≤n, then we have 1. limr→∞Ir(a) = limr→∞I_r⁰(a) =I(a),

2. limr→∞L_r(a) = limr→∞L⁰_r(a) = L(a).

Proof. It is easy to see thatlimr→∞I_r(a) = limr→∞I_r⁰(a)andlimr→∞L_r(a) = lim_r→∞L⁰_r(a), since

r→∞lim π¹/(^2n+r−2n+r−1)

j = 1, lim

r→∞

1

2n+r−2 n+r−1

n−1

X

j=1

δ_j = 0, lim

r→∞

n+r−2 r−1

2n+r−2 n+r−1

= 1.

Straightforward computation yields ln lim

r→∞I_r⁰(a)

= lim

r→∞lnI_r⁰(a)

= lim

r→∞

1

n+r−1 r

X

i1+i2+···+in=r i1,i2,...,in≥0

ln

n

X

k=1

ik

ra_k

= (n−1)!

Z

· · · Z

x1+x2+···+xn−1≤1 x`≥0,1≤`≤n−1

ln

"

1−

n−1

X

i=1

x_i

! a₁+

n

X

j=2

xj−1a_j

#

dx₁dx₂. . . dxn−1

= (n−1)!

V(a)

n

X

i=1

(−1)ⁿ⁺ⁱaⁿ⁻¹_i V_i(a)(lna_i−m) = lnI(a)

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and

r→∞lim L_r(a) (3.11)

= lim

r→∞

1

n+r−1 r

X

i1+i2+···+in=r i1,i2,...,in≥0

n

Y

k=1

aⁱ_k^k^/r

= (n−1)!

Z

· · · Z

x1+x2+···+xn−1≤1 x`≥0,1≤`≤n−1

a¹⁻

Pn−1 i=1 xi

1 a^x₂¹· · ·a^x_nⁿ⁻¹dx₁dx₂· · ·dxn−1

= (n−1)!

V(lna)

n

X

i=1

(−1)ⁿ⁺ⁱaiVi(lna) =L(a).

The proof is complete.

Proof of Theorem2.1. This follows from combination of Lemma3.1and Lemma3.2.

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References

[1] E. LEACH AND M. SHOLANDER, Extended mean values, Amer. Math.

Monthly, 85 (1978), 84–90.

[2] A.O. PITTENGER, Two logarithmic mean in n variables, Amer. Math.

Monthly, 92 (1985), 99–104.

[3] G. PÓLYA AND G. SZEGÖ, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951.

[4] K.B. STOLARSKY, Generalizations of the logarithmic mean, Mag. Math., 48 (1975), 87–92.