Journal of Inequalities in Pure and
Applied Mathematics
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
D EPARTMENT OF M ATHEMATICS ,
H UNAN I NSTITUTE OF S CIENCE AND T ECHNOLOGY,
Y UEYANG C ITY, H UNAN 414006,
CHINA.
xzgzzh@163.com
Z IXING E DUCATIONAL R ESEARCH S ECTION ,
C HENZHOU C ITY,
H UNAN 423400, CHINA.
zxzh1234@163.com
Received 21 October, 2002; accepted 28 March, 2003
Communicated by F. Qi
A BSTRACT. In the short note, the inequalities G ≤ L ≤ I ≤ A for the geometric, logarithmic,
identric, and arithmetic means in n variables are proved.
Key words and phrases: Inequality, Arithmetic mean, Geometric mean, Logarithmic mean, Identric mean, n variables, Van
der Monde determinant.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
For positive numbers ai , 1 ≤ i ≤ 2, let
a1 + a2
(1.1)
A = A(a1 , a2 ) =
;
√ 2
(1.2)
G = G(a1 , a2 ) = a1 a2 ;

exp a2 ln a2 − a1 ln a1 − 1 , a < a ,
1
2
a2 − a1
(1.3)
I = I(a1 , a2 ) =

a1 ,
a 1 = a2 ;

 a2 − a1 , a < a ,
1
2
L = L(a1 , a2 ) = ln a2 − ln a1
(1.4)
a ,
a =a .
1
1
2
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
2
Z H .-G. X IAO AND Z H .-H. Z HANG
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.
2. D EFINITIONS AND THE M AIN R ESULT
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 ,
#
"
n
1 X
n+i n−1
I = I(a) = I(a1 , . . . , an ) = exp
(−1) ai Vi (a) ln ai − m ,
V (a) i=1
A = A(a) = A(a1 , . . . , an ) =
n
(2.4)
L = L(a) = L(a1 , . . . , an ) =
(n − 1)! X
(−1)n+i ai Vi (ln a),
V (ln a) i=1
where ln a = (ln a1 , . . . , ln an ), ai 6= aj for i 6= j,
(2.5)
1
1
a1
a
2
a22
V (a) = a21
...
..
n−1 .n−1
a1
a2
...
1 . . . an Y
. . . a2n =
(ai − aj )
. . . . . . 1≤j<i≤n
. . . an−1
n
is the determinant of Van der Monde’s matrix of the n-th order,
(2.6)
1
1
a1
a
2
a22
Vi (a) = a21
...
..
n−2 .n−2
a1
a2
...
1
1
. . . ai−1 ai+1
. . . a2i−1 a2i+1
... ...
...
n−2
. . . ai−1 an−2
i+1
...
1 . . . an . . . a2n ,
. . . . . . . . . an−2
n
P
1
m = n−1
k=1 k , and 1 ≤ i ≤ n.
The main result of this short note can be stated as
Theorem 2.1. Let a = (a1 , a2 , . . . , an ) and ai > 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 if a1 = a2 = · · · = an .
J. Inequal. Pure and Appl. Math., 4(2) Art. 39, 2003
http://jipam.vu.edu.au/
T HE I NEQUALITIES G ≤ L ≤ I ≤ A IN n VARIABLES
3
3. P ROOF OF T HEOREM 2.1
To prove inequalities in (2.7), we introduce the following means
1
# n+r−2
" n
X
Y
( r−1 )
ik
(3.1)
,
ak
Ir (a) =
n+r−1
i +i +···+i =n+r−1 k=1
1
n
2
i1 ,i2 ,...,in ≥1
"
Y
Ir0 (a) =
(3.2)
i1 +i2 +···+in =r
i1 ,i2 ,...,in ≥0
Lr (a) =
(3.3)
1
L0r (a) =
k=1
r
X
#
ak
1
(n+r−1
)
r
n
Y
n+r−1
r
(3.4)
n
X
ik
,
i /r
akk ,
i1 +i2 +···+in =r k=1
i1 ,i2 ,...,in ≥0
1
n
Y
X
n+r−2
r−1
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
(1) I1 (a) = L1 (a) = A(a);
(2) I10 (a) = L01 (a) = G(a);
(3) For 1 ≤ j ≤ n − 1 and
n
Y
X
ik
(3.5)
πj =
ak ,
n+r−1
i +i +···+i =n+r−1
k=1
1
n
2
ik1 =ik2 =···=ikj =0, for the rest ik ≥1
we have
0
In+r−1
(a)
(3.6)
=
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
(4) For 1 ≤ j ≤ n − 1 and
n
Y
X
δj =
(3.7)
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
1
" n−1
X
(5) If r ∈ N, then
(a) Ir (a) ≥ Ir+1 (a),
0
(b) Ir0 (a) ≤ Ir+1
(a),
(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 .
J. Inequal. Pure and Appl. Math., 4(2) Art. 39, 2003
http://jipam.vu.edu.au/
4
Z H .-G. X IAO AND Z H .-H. Z HANG
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
n
X ik
ik
n+r
ak =
·
ak
n+r−1
n
+
r
n
+
r
−
1
k=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 ik
X
ik
1
ij
ak +
ak
=
n + r − 1 j=1 k=1 n + r
n+r
k=1
" n
#
n
n
X X
X
ik
ij
1
ij
ak +
aj
=
n + r − 1 j=1 k=1 n + r
n
+
r
j=1
" n
#
n
X i k ak
X
aj
ij
+
.
=
n + r − 1 k=1 n + r n + r
j=1
By using the weighted arithmetic-geometric mean inequality, we have
n
X
k=1
" n
#ij /(n+r−1)
n
Y
X ik a k
ik
aj
ak ≥
+
,
n+r−1
n+r n+r
j=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
≥
" n
#ij /(n+r−1)
n
Y
X ik a k
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
J. Inequal. Pure and Appl. Math., 4(2) Art. 39, 2003
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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T HE I NEQUALITIES G ≤ L ≤ I ≤ A IN n VARIABLES
"
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
5
# (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
X
νk
ak
n+r
k=1
#(νj −1)/(n+r−1)
= 1 for νj = 1.
The equalities above are valid if and only if
n
n
n
X
X
X
ik ak
a1
ik a k
a2
i k ak
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
"
Y
(3.10)
i1 +i2 +···+in =r+1
i1 ≥1,i2 ≥1,...,in ≥1
n
X
# r1
"
X
≥
i j aj
j=1
i1 +i2 +···+in =r
i1 ≥0,i2 ≥0,...,in ≥0
n
Y
# r1
i
ajj
.
j=1
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
lim
r→∞
1/(2n+r−2)
πj n+r−1
= 1,
J. Inequal. Pure and Appl. Math., 4(2) Art. 39, 2003
lim
r→∞
n−1
X
1
2n+r−2
n+r−1
j=1
n+r−2
r−1
2n+r−2
n+r−1
δj = 0,
lim
r→∞
= 1.
http://jipam.vu.edu.au/
6
Z H .-G. X IAO AND Z H .-H. Z HANG
Straightforward computation yields
ln lim Ir0 (a) = lim ln Ir0 (a)
r→∞
r→∞
= lim
r→∞
1
X
n+r−1
r
i1 +i2 +···+in =r
i1 ,i2 ,...,in ≥0
Z
= (n − 1)!
=
ln
n
X
ik
k=1
"
Z
···
ln
1−
r
ak
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
m)
= ln I(a)
and
(3.11)
lim Lr (a) = lim
r→∞
r→∞
1
X
n+r−1
r
Z
= (n − 1)!
=
n
Y
i /r
akk
i1 +i2 +···+in =r k=1
i1 ,i2 ,...,in ≥0
Z
P
1− n−1 x
···
a1 i=1 i ax2 1
· · · axnn−1 dx1 dx2 · · · dxn−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=1
= L(a).
The proof is complete.
Proof of Theorem 2.1. This follows from combination of Lemma 3.1 and Lemma 3.2.
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 4(2) Art. 39, 2003
http://jipam.vu.edu.au/