Journal of Inequalities in Pure and
Applied Mathematics
THE INEQUALITIES G ≤ L ≤ I ≤ A IN n 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
volume 4, issue 2, article 39,
2003.
Received 21 October, 2002;
accepted 28 March, 2003.
Communicated by: F. Qi
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
110-02
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Abstract
In the short note, the inequalities G ≤ L ≤ I ≤ A for the geometric, logarithmic,
identric, and arithmetic means in n variables are proved.
2000 Mathematics Subject Classification: 26D15.
Key words: Inequality, Arithmetic mean, Geometric mean, Logarithmic mean, Identric mean, n variables, Van der Monde determinant.
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
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.
Zhen-Gang Xiao and
Zhi-Hua Zhang
Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Definitions and the Main Result . . . . . . . . . . . . . . . . . . . . . . . .
Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3
4
6
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1.
Introduction
For positive numbers ai , 1 ≤ i ≤ 2, let
(1.1)
(1.2)
(1.3)
(1.4)
a1 + a2
;
√ 2
G = G(a1 , a2 ) = a1 a2 ;

exp a2 ln a2 − a1 ln a1 − 1 , a < a ,
1
2
a2 − a1
I = I(a1 , a2 ) =

a1 ,
a1 = a2 ;

 a2 − a1 , a < a ,
1
2
L = L(a1 , a2 ) = ln a2 − ln a1
a ,
a 1 = a2 .
1
A = A(a1 , a2 ) =
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 mean I
and proved G ≤ L ≤ I ≤ A in two variables in [4]. See [2] also.
The purpose of this short note is to prove the inequalities G ≤ L ≤ I ≤ A
of the geometric, logarithmic, identric, and arithmetic means in n variables.
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
Zhen-Gang Xiao and
Zhi-Hua Zhang
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2.
Definitions and the Main Result
Let a = (a1 , a2 , . . . , an ) and ai > 0 for 1 ≤ i ≤ n, then the arithmetic, geometric, identric, and logarithmic means in n variables are defined respectively
as follows
(2.1)
(2.2)
(2.3)
a1 + a2 + · · · + an
,
n
√
G = G(a) = G(a1 , . . . , an ) = n a1 a2 · · · an ,
I = I(a) = I(a1 , . . . , an )
"
#
n
X
1
= exp
(−1)n+i an−1
Vi (a) ln ai − m ,
i
V (a) i=1
A = A(a) = A(a1 , . . . , an ) =
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
Zhen-Gang Xiao and
Zhi-Hua Zhang
n
(2.4)
(n − 1)! X
(−1)n+i ai Vi (ln a),
L = L(a) = L(a1 , . . . , an ) =
V (ln a) i=1
where ln a = (ln a1 , . . . , ln an ), ai
1
1
a1
a2
a22
(2.5)
V (a) = a21
...
..
n−1 .n−1
a
a
1
2
6= aj for i 6= j,
...
1 . . . an Y
. . . a2n =
(ai − aj )
. . . . . . 1≤j<i≤n
. . . an−1 n
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is the determinant of Van der Monde’s matrix of the n-th order,
1
1
...
1
1
...
1 a1
a2 . . . ai−1 ai+1 . . . an 2
a22 . . . a2i−1 a2i+1 . . . a2n ,
(2.6)
Vi (a) = a1
...
.. ... ...
. . . . . . . . . n−2 .n−2
n−2
a
a2
. . . ai−1 an−2
. . . ann−2 1
i+1
Pn−1 1
, and 1 ≤ i ≤ n.
m = k=1
k
The main result of this short note can be stated as
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
Theorem 2.1. Let a = (a1 , a2 , . . . , an ) and ai > 0 for 1 ≤ i ≤ n, then the
inequalities
Zhen-Gang Xiao and
Zhi-Hua Zhang
G(a) ≤ L(a) ≤ I(a) ≤ A(a)
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of the geometric, logarithmic, identric, and arithmetic means in n variables
hold. The equalities in (2.7) are valid if and only if a1 = a2 = · · · = an .
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(2.7)
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3.
Proof of Theorem 2.1
To prove inequalities in (2.7), we introduce the following means
"
(3.1)
Y
Ir (a) =
n
X
i1 +i2 +···+in =n+r−1
i1 ,i2 ,...,in ≥1
"
(3.2)
Ir0 (a) =
Y
i1 +i2 +···+in =r
i1 ,i2 ,...,in ≥0
(3.3)
(3.4)
Lr (a) =
L0r (a) =
1
n+r−1
r
n
X
ik
k=1
r
X
#
ak
1
(n+r−1
)
r
n
Y
,
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
Zhen-Gang Xiao and
Zhi-Hua Zhang
i /r
akk ,
i1 +i2 +···+in =r k=1
i1 ,i2 ,...,in ≥0
1
n+r−2
r−1
k=1
1
# n+r−2
( r−1 )
ik
ak
,
n+r−1
X
n
Y
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i /(n+r−1)
akk
,
i1 +i2 +···+in =n+r−1 k=1
i1 ,i2 ,...,in ≥1
where a = (a1 , a2 , . . . , an ) and ai > 0 for 1 ≤ i ≤ n.
Lemma 3.1. Let a = (a1 , a2 , . . . , an ) and ai > 0 for 1 ≤ i ≤ n, then we have
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1. I1 (a) = L1 (a) = A(a);
2. I10 (a) = L01 (a) = G(a);
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3. For 1 ≤ j ≤ n − 1 and
(3.5)
n
X
Y
πj =
i1 +i2 +···+in =n+r−1
k=1
ik1 =ik2 =···=ikj =0, for the rest ik ≥1
ik
ak ,
n+r−1
we have
(3.6)
0
In+r−1
(a)
=
n−1
Y
1
πj
2n+r−2
(n+r−2
(2n+r−2
n+r−1 )
r−1 ) ( n+r−1 )
Ir (a)
;
j=1
Zhen-Gang Xiao and
Zhi-Hua Zhang
4. For 1 ≤ j ≤ n − 1 and
(3.7)
n
Y
X
δj =
i /(n+r−1)
akk
,
i1 +i2 +···+in =n+r−1
k=1
ik1 =ik2 =···=ikj =0, for the rest ik ≥1
we have
(3.8)
#
n+r−2 0
Ln+r−1 (a) = 2n+r−2
δj +
Lr (a) ;
r
−
1
n+r−1
j=1
5. If r ∈ N, then
(a) Ir (a) ≥ Ir+1 (a),
0
(b) Ir0 (a) ≤ Ir+1
(a),
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
1
" n−1
X
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(c) Lr (a) ≥ Lr+1 (a),
(d) L0r (a) ≤ L0r+1 (a),
(e) Ir (a) ≥ Lr (a),
(f) Ir0 (a) ≥ L0r (a),
where equalities above hold if and only if a1 = a2 · · · = an .
Proof. The formula (3.6) follows from standard arguments and formulas (3.1)
and (3.2).
If r ∈ N and i1 + i2 + · · · + in = n + r − 1, then
n
X
k=1
ik
ak =
n+r−1
n
X
k=1
ik
n+r
·
ak
n+r n+r−1
n
=
=
=
=
n + r X ik
ak
n + r − 1 k=1 n + r
" n
# n
X
X ik
1
ij + 1
ak
n + r − 1 j=1
n+r
k=1
" n
#
n
n
X X
X
ik
ik
1
ij
ak +
ak
n + r − 1 j=1 k=1 n + r
n
+
r
k=1
" n
#
n
n
X X ik
X
ij
1
ij
ak +
aj
n + r − 1 j=1 k=1 n + r
n+r
j=1
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
Zhen-Gang Xiao and
Zhi-Hua Zhang
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=
n
X
j=1
" n
#
X ik a k
ij
aj
.
+
n + r − 1 k=1 n + r n + r
By using the weighted arithmetic-geometric mean inequality, we have
#ij /(n+r−1)
" n
n
n
X
Y
X ik a k
ik
aj
,
ak ≥
+
n+r−1
n+r n+r
j=1 k=1
k=1
and then
(3.9)
n
X
Y
i1 +i2 +···+in =n+r−1 k=1
i1 ,i2 ,...,in ≥1
ik
ak
n+r−1
Y
≥
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
Zhen-Gang Xiao and
Zhi-Hua Zhang
#ij /(n+r−1)
" n
n
Y
X i k ak
aj
+
n+r n+r
=n+r−1 j=1 k=1
i1 +i2 +···+in
i1 ,i2 ,...,in ≥1
=
"
n
Y
Y
j=1 i1 +i2 +···+in =n+r−1
i1 ,i2 ,...,in ≥1
=
"
n
Y
j=1
Y
ν1 +ν2 +···+νn =n+r
ν1 ,ν2 ,...,νj−1 ,νj+1 ,...,νn ≥1;νj ≥2
"
=
n
X
i k ak
aj
+
n+r n+r
k=1
Y
ν1 +ν2 +···+νn =n+r
ν1 ,ν2 ,...,νn ≥1
n
X
νk
ak
n+r
k=1
#ij /(n+r−1)
n
X
νk
ak
n
+
r
k=1
#(νj −1)/(n+r−1)
#Pnj=1 (νj −1)/(n+r−1)
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"
=
Y
ν1 +ν2 +···+νn =n+r
ν1 ,ν2 ,...,νn ≥1
"
=
Y
ν1 +ν2 +···+νn =n+r
ν1 ,ν2 ,...,νn ≥1
"
=
Y
ν1 +ν2 +···+νn =n+r
ν1 ,ν2 ,...,νn ≥1
# (Pnj=1 νj −n)/(n+r−1)
n
X
νk
ak
n
+
r
k=1
n
X
νk
ak
n+r
k=1
#r/(n+r−1)
n+r−1
# (n+r−2
)
n
r−1 )/(
r
X
νk
ak
,
n
+
r
k=1
notice that the result from line 4 to line 5 in (3.9) follows from a simple fact that
" n
#(νj −1)/(n+r−1)
X νk
ak
= 1 for νj = 1.
n
+
r
k=1
The equalities above are valid if and only if
n
n
n
X
X
X
ik a k
a1
i k ak
a2
ik a k
an
+
=
+
= ··· =
+
,
n+r n+r
n+r n+r
n+r n+r
k=1
k=1
k=1
which is equivalent to a1 = a2 = · · · = an . This implies that Ir (a) ≥ Ir+1 (a).
The inequality Ir (a) ≥ Lr (a) follows easily from the generalized Hölder
inequality
" n
#1
" n
# r1
X
X
Y i r
Y
(3.10)
i j aj ≥
ajj .
i1 +i2 +···+in =r+1
i1 ≥1,i2 ≥1,...,in ≥1
j=1
i1 +i2 +···+in =r
i1 ≥0,i2 ≥0,...,in ≥0
j=1
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
Zhen-Gang Xiao and
Zhi-Hua Zhang
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The proofs of other formulas and inequalities will be left to the readers.
Lemma 3.2. Let a = (a1 , a2 , . . . , an ) and ai > 0 for 1 ≤ i ≤ n, then we have
1. limr→∞ Ir (a) = limr→∞ Ir0 (a) = I(a),
2. limr→∞ Lr (a) = limr→∞ L0r (a) = L(a).
Proof. It is easy to see that limr→∞ Ir (a) = limr→∞ Ir0 (a) and limr→∞ Lr (a) =
limr→∞ L0r (a), since
n−1
n+r−2
X
1/(2n+r−2
1
r−1
n+r−1 )
lim πj
= 1, lim 2n+r−2
δj = 0, lim 2n+r−2 = 1.
r→∞
r→∞
n+r−1
r→∞
j=1
n+r−1
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
Zhen-Gang Xiao and
Zhi-Hua Zhang
Straightforward computation yields
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ln lim Ir0 (a)
r→∞
Contents
= lim ln Ir0 (a)
r→∞
= lim
r→∞
1
X
n+r−1
r
i1 +i2 +···+in =r
i1 ,i2 ,...,in ≥0
Z
= (n − 1)!
=
ln
k=1
"
Z
···
n
X
ik
ln
1−
r
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n−1
X
!
x i a1 +
i=1
x1 +x2 +···+xn−1 ≤1
x` ≥0,1≤`≤n−1
n
(n − 1)! X
(−1)n+i an−1
Vi (a)(ln ai −
i
V (a) i=1
n
X
#
xj−1 aj dx1 dx2 . . . dxn−1
j=2
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m) = ln I(a)
J. Ineq. Pure and Appl. Math. 4(2) Art. 39, 2003
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and
(3.11)
lim Lr (a)
r→∞
1
= lim n+r−1
r→∞
r
Z
= (n − 1)!
=
X
n
Y
i1 +i2 +···+in =r k=1
i1 ,i2 ,...,in ≥0
Z
P
1− n−1 x
a1 i=1 i ax2 1
···
x1 +x2 +···+xn−1 ≤1
x` ≥0,1≤`≤n−1
n
1)! X
(−1)n+i ai Vi (ln a)
(n −
V (ln a)
i /r
akk
· · · axnn−1 dx1 dx2 · · · dxn−1
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
= L(a).
Zhen-Gang Xiao and
Zhi-Hua Zhang
i=1
The proof is complete.
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Proof of Theorem 2.1. This follows from combination of Lemma 3.1 and Lemma 3.2.
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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.
The Inequalities G ≤ L ≤ I ≤ A
in n Variables
Zhen-Gang Xiao and
Zhi-Hua Zhang
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