INTRODUCTION For positive numbersai,1≤i≤2, let A=A(a1, a2

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

Volume 4, Issue 2, Article 39, 2003

THE INEQUALITIES G≤L≤I ≤A IN n VARIABLES

ZHEN-GANG XIAO AND ZHI-HUA ZHANG DEPARTMENT OFMATHEMATICS,

HUNANINSTITUTE OFSCIENCE ANDTECHNOLOGY, YUEYANGCITY, HUNAN414006,

CHINA.

xzgzzh@163.com

ZIXINGEDUCATIONALRESEARCHSECTION, CHENZHOUCITY,

HUNAN423400, CHINA.

zxzh1234@163.com

Received 21 October, 2002; accepted 28 March, 2003 Communicated by F. Qi

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

Key words and phrases: Inequality, Arithmetic mean, Geometric mean, Logarithmic mean, Identric mean,nvariables, Van der Monde determinant.

2000 Mathematics Subject Classification. 26D15.

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.

ISSN (electronic): 1443-5756 c

2003 Victoria University. All rights reserved.

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.

110-02

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

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

2. DEFINITIONS AND THEMAIN RESULT

Leta = (a₁, a₂, . . . , a_n)anda_i >0for1 ≤ i ≤ n, then the arithmetic, geometric, identric, and logarithmic means innvariables are defined respectively as follows

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

n ,

(2.1)

G=G(a) = G(a1, . . . , an) = √ⁿ

a1a2· · ·an, (2.2)

I =I(a) = I(a₁, . . . , a_n) = exp

"

1 V(a)

n

X

i=1

(−1)ⁿ⁺ⁱaⁿ⁻¹_i V_i(a) lna_i−m

# , (2.3)

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)

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

(2.6) V_i(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. Leta= (a₁, a₂, . . . , a_n)anda_i >0for1≤i≤n, then the inequalities

(2.7) G(a)≤L(a)≤I(a)≤A(a)

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

3. PROOF OFTHEOREM2.1 To prove inequalities in (2.7), we introduce the following means

I_r(a) = Y

i1+i2+···+in=n+r−1 i1,i2,...,in≥1

" _n X

k=1

i_k n+r−1a_k

# ¹ (^n+r−2r−1 )

, (3.1)

I_r⁰(a) = Y

i1+i2+···+in=r i1,i2,...,in≥0

" _n X

k=1

i_k ra_k

# ¹ (^n+r−1r )

, (3.2)

L_r(a) = 1

n+r−1 r

X

i1+i2+···+i_n=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, . . . , a_n)anda_i >0for1≤i≤n, then we have (1) I₁(a) = L₁(a) =A(a);

(2) I₁⁰(a) = L⁰₁(a) =G(a);

(3) For1≤j ≤n−1and

(3.5) π_j = Y

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

i_k n+rak

= 1

n+r−1

" _n X

j=1

ij + 1

# _n X

k=1

i_k n+rak

= 1

n+r−1

" _n X

j=1

ij n

X

k=1

i_k

n+rak+

n

X

k=1

i_k n+rak

#

= 1

n+r−1

" _n X

j=1

ij n

X

k=1

i_k

n+rak+

n

X

j=1

i_j n+raj

#

=

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

ik

n+r−1ak

(3.9)

≥ Y

i1+i2+···+i_n=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+ra_k

#^Pn_j=1(νj−1)/(n+r−1)

= Y

ν1+ν2+···+ν_n=n+r ν1,ν2,...,νn≥1

" _n X

k=1

ν_k n+ra_k

#(^Pnj=1νj−n)/^(n+r−1)

= Y

ν1+ν2+···+ν_n=n+r ν1,ν2,...,νn≥1

" _n X

k=1

νk

n+ra_k

#r/(n+r−1)

= Y

ν1+ν2+···+νn=n+r ν1,ν2,...,νn≥1

" _n X

k=1

ν_k 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

νk

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

(3.10) Y

i1+i2+···+in=r+1 i1≥1,i2≥1,...,in≥1

" _n X

j=1

i_ja_j

#¹_r

≥ X

i1+i2+···+in=r i1≥0,i2≥0,...,in≥0

" _n Y

j=1

aⁱ_j^j

#¹_r .

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→∞I_r(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→∞Ir(a) = limr→∞I_r⁰(a)andlimr→∞Lr(a) = limr→∞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+···+i_n=r i1,i2,...,in≥0

ln

n

X

k=1

i_k 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) and

r→∞lim L_r(a) = lim

r→∞

1

n+r−1 r

X

i1+i2+···+in=r i1,i2,...,in≥0

n

Y

k=1

aⁱ_k^k/r (3.11)

= (n−1)!

Z

· · · Z

x1+x2+···+xn−1≤1 x<sub>`</sub>≥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⁺ⁱa_iV_i(lna)

=L(a).

The proof is complete.

Proof of Theorem 2.1. This follows from combination of Lemma 3.1 and Lemma 3.2.

REFERENCES

[1] E. LEACHANDM. SHOLANDER, Extended mean values, Amer. Math. Monthly, 85 (1978), 84–90.

[2] A.O. PITTENGER, Two logarithmic mean innvariables, Amer. Math. Monthly, 92 (1985), 99–104.

[3] G. PÓLYAANDG. SZEGÖ, Isoperimetric Inequalities in Mathematical Physics, Princeton Univer- sity Press, Princeton, 1951.

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