Journal of Inequalities in Pure and
Applied Mathematics
EXTENDING MEANS OF TWO VARIABLES TO SEVERAL
VARIABLES
volume 5, issue 3, article 65,
2004.
JORMA K. MERIKOSKI
Department of Mathematics, Statistics and Philosophy
FIN-33014 University of Tampere
Finland.
EMail: jorma.merikoski@uta.fi
Received 22 January, 2004;
accepted 10 June, 2004.
Communicated by: S. Puntanen
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
We present a method, based on series expansions and symmetric polynomials,
by which a mean of two variables can be extended to several variables. We
apply it mainly to the logarithmic mean.
2000 Mathematics Subject Classification: 26E60, 26D15.
Key words: Means, Logarithmic mean, Divided differences, Series expansions,
Symmetric polynomials.
Extending Means of Two
Variables to Several Variables
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Polynomials Corresponding to a Mean . . . . . . . . . . . . . . . . . . .
3
Trivial Extensions: A and G . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Extending L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Numerical Computation of L . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Inequality G ≤ L ≤ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Inequalities Pm [G] ≤ Pm [L] ≤ Pm [A] . . . . . . . . . . . . . . . . . .
8
Other Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Jorma K. Merikoski
3
5
9
11
15
16
18
19
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1.
Introduction
Throughout this paper, n ≥ 2 is an integer and x1 , . . . , xn are positive real
numbers.
The logarithmic mean of x1 and x2 is defined by
(1.1)
x1 − x2
ln x1 − ln x2
L(x1 , x1 ) = x1 .
if x1 6= x2 ,
L(x1 , x2 ) =
There are several ways to extend this to n variables. Bullen ([1, p. 391]) writes
that perhaps the most natural extension is due to Pittenger [13]. Based on an
integral, it is
"
#−1
n
X
xn−2
ln xi
i
Qn
(1.2)
L(x1 , . . . , xn ) = (n − 1)
j=1 (ln xi − ln xj )
i=1
j6=i
if all the xi ’s are unequal. Bullen ([1, p. 392]) also writes that another natural
extension has been given by Neuman [9]. Based on the integral (6.3), it is
(1.3)
L(x1 , . . . , xn ) = (n − 1)!
n
X
i=1
Extending Means of Two
Variables to Several Variables
Jorma K. Merikoski
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xi
Qn
j=1 (ln xi − ln xj )
j6=i
if all the xi ’s are unequal. It is obviously different from (1.2).
If some of the xi ’s are equal, then (1.2) and (1.3) are defined by continuity.
Mustonen [6] gave (1.3) in 1976 but published it only recently [7] in the
home page of his statistical data processing system, not in a journal. We will
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present his method. It is based on a series expansion and supports the notion
that (1.3) is the most natural extension of (1.1).
In general, we call a continuous real function µ of two positive (or nonnegative) variables a mean if, for all x1 , x2 , c > 0 (or x1 , x2 , c ≥ 0),
(i1 ) µ(x1 , x2 ) = µ(x2 , x1 ),
(i2 ) µ(x1 , x1 ) = x1 ,
(i3 ) µ(cx1 , cx2 ) = cµ(x1 , x2 ),
(i4 ) x1 ≤ y1 , x2 ≤ y2 ⇒ µ(x1 , x2 ) ≤ µ(y1 , y2 ),
Extending Means of Two
Variables to Several Variables
Jorma K. Merikoski
(i5 ) min(x1 , x2 ) ≤ µ(x1 , x2 ) ≤ max(x1 , x2 ).
Axiomatization of means is widely studied, see e.g. [1] and references
therein.
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2.
Polynomials Corresponding to a Mean
To extend the arithmetic and geometric means
1
x1 + x2
A(x1 , x2 ) =
, G(x1 , x2 ) = (x1 x2 ) 2
2
to n variables is trivial, but to visualize our method, it may be instructive.
Substituting
x1 = eu1 , x2 = eu2 ,
(2.1)
we have
(2.2)
A(x1 , x2 ) = Ã(u1 , u2 )
1
= (eu1 + eu2 )
2
1
u21
u22
=
1 + u1 +
+ · · · + 1 + u2 +
+ ···
2
2!
2!
u1 + u 2
1 u2 + u22
1 u3 + u32
=1+
+ · 1
+ · 1
+ ··· ,
2
2!
2
3!
2
Extending Means of Two
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Jorma K. Merikoski
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(2.3) G(x1 , x2 ) = G̃(u1 , u2 )
1
= (eu1 eu2 ) 2 = e
u1 +u2
2
2
u1 + u2
1 u1 + u2
=1+
+
+ ···
2
2!
2
1 (u1 + u2 )2
1 (u1 + u2 )3
u1 + u2
=1+
+ ·
+
·
+ ··· ,
2
2!
22
3!
23
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(2.4) L(x1 , x2 ) = L̃(u1 , u2 )
eu1 − eu2
=
u − u2
1
u21
u22
= 1 + u1 +
+ · · · − 1 − u2 −
− · · · (u1 − u2 )−1
2!
2!
u21 − u22 u31 − u32
= u 1 − u2 +
+
+ · · · (u1 − u2 )−1
2!
3!
2
u1 + u 2
1 u + u1 u2 + u22
=1+
+ · 1
2
2!
3
1 u31 + u21 u2 + u1 u22 + u32
+ ·
+ ··· .
3!
4
All these expansions are of the form
1
1
P2 (u1 , u2 ) + P3 (u1 , u2 ) + · · · ,
2!
3!
where the Pm ’s are symmetric homogeneous polynomials of degree m. In all of
them,
u1 + u2
= A(u1 , u2 ).
P1 (u1 , u2 ) =
2
The coefficients of
(2.5)
(2.6)
1 + P1 (u1 , u2 ) +
m−1
Pm (u1 , u2 ) = b0 um
u 2 + · · · + bm um
1 + b1 u1
2
Extending Means of Two
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Jorma K. Merikoski
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are nonnegative numbers with sum 1. They are for A
1
1
b0 = , b1 = · · · = bm−1 = 0, bm = ,
2
2
J. Ineq. Pure and Appl. Math. 5(3) Art. 65, 2004
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for G
m −m
2
bk =
k
(0 ≤ k ≤ m),
and for L
1
.
m+1
Let µ be a mean of two variables. Assume that it has a valid expansion (2.5).
Fix m ≥ 2, and denote by Pm [µ] the polynomial (2.6). Its coefficients define
a discrete random variable, denoted by Xm [µ], whose value is k (0 ≤ k ≤ m)
with probability bk . In particular, Xm [A] is distributed uniformly over {0, m},
and Xm [G] binomially and Xm [L] uniformly over {0, . . . , m}. Their variances
satisfy
Var Xm [G] ≤ Var Xm [L] ≤ Var Xm [A],
b0 = · · · = bm =
which is an interesting reminiscent of
(2.7)
Pm [G](u1 , u2 ) ≤ Pm [L](u1 , u2 ) ≤ Pm [A](u1 , u2 )
for all m ≥ 1. The functions
Rm [µ](u1 , u2 ) = (Pm [µ](u1 , u2 ))
Jorma K. Merikoski
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G(x1 , x2 ) ≤ L(x1 , x2 ) ≤ A(x1 , x2 ).
Let u1 , u2 ≥ 0, then (2.7) holds in fact termwise:
(2.8)
Extending Means of Two
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1
m
are means. In particular, for A they are moment means
m
m1
u 1 + um
2
Rm [A](u1 , u2 ) =
= Mm (u1 , u2 ),
2
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for G all of them are equal to the arithmetic mean
Rm [G](u1 , u2 ) =
u1 + u2
= A(u1 , u2 ),
2
and for L they are special cases of complete symmetric polynomial means and
Stolarsky means (see e.g. [1, pp. 341, 393])
um+1
− um+1
1
2
Rm [L](u1 , u2 ) =
(m + 1)(u1 − u2 )
m1
=
m−1
um
u2 + · · · + u m
1 + u1
2
m+1
m1
.
Since the Pm |µ]’s are symmetric and homogeneous polynomials of two variables, they can be extended to n variables. Thus µ can also be likewise extended.
Extending Means of Two
Variables to Several Variables
Jorma K. Merikoski
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3.
Trivial Extensions: A and G
Consider first A. By (2.2),
m
um
1 + u2
.
2
Pm [A](u1 , u2 ) =
To extend it to n variables is actually as trivial as to extend A directly. We obtain
Pm [A](u1 , . . . , un ) =
m
um
1 + · · · + un
,
n
Extending Means of Two
Variables to Several Variables
and so
Jorma K. Merikoski
∞
X
1
A(x1 , . . . , xn ) =
Pm [A](u1 , . . . , un )
m!
m=0
=
1
n
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∞
∞
X
X
um
um
1
n
+ ··· +
m!
m!
m=0
m=0
!
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1
x1 + · · · + xn
= (eu1 + · · · + eun ) =
.
n
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Next, study G. By (2.3),
Pm [G](u1 , u2 ) =
u1 + u2
2
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which can be immediately extended to
Pm [G](u1 , . . . , un ) =
JJ
J
u1 + · · · + u n
n
m
,
J. Ineq. Pure and Appl. Math. 5(3) Art. 65, 2004
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and so
∞
X
1
G(x1 , . . . , xn ) =
Pm [G](u1 , . . . , un )
m!
m=0
m
∞
X
1
u1 + · · · + u n
=
m!
n
m=0
=e
u1 +···+un
n
1
1
= (eu1 · · · eun ) n = (x1 · · · xn ) n .
We present a “termwise” (cf. (2.8)) proof of the geometric-arithmetic mean
inequality
(3.1)
G(x1 , . . . , xn ) ≤ A(x1 , . . . , xn ).
Pm [G](u1 , . . . , un ) ≤ Pm [A](u1 , . . . , un )
or equivalently
(3.3)
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We can assume that u1 , . . . , un ≥ 0; if not, consider cG ≤ cA for a suitable
c > 0. Let m ≥ 1. Then
(3.2)
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Rm [G](u1 , . . . , un ) ≤ Rm [A](u1 , . . . , un ),
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since
u1 + · · · + u n
≤
n
m
um
1 + · · · + un
n
1
m
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J. Ineq. Pure and Appl. Math. 5(3) Art. 65, 2004
by Schlömilch’s inequality (see e.g. [1, p. 203]). Therefore (3.1) follows.
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4.
Extending L
Let 1 ≤ m ≤ n. The mth complete symmetric polynomial of u1 , . . . , un ≥ 0
(see e.g. [1, p. 341]) is defined by
X
Cm (u1 , . . . , un ) =
u1 i1 · · · un in .
i1 +···+in =m
(Here i1 , . . . , in ≥ 0, and we define 00 = 1.)
Let us now study L. Denote Qm = Pm [L]. By (2.4),
Extending Means of Two
Variables to Several Variables
m−1
um
u2 + · · · + u m
1 + u1
2
Qm (u1 , u2 ) =
.
m+1
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This can be easily extended to
(4.1)
Qm (u1 , . . . , un ) =
Jorma K. Merikoski
−1
n+m−1
Cm (u1 , . . . , un ).
m
The corresponding mean,
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1
m
Rm [L](u1 , . . . , un ) = Qm (u1 , . . . , un ) ,
is called [1] the mth complete symmetric polynomial mean of u1 , . . . , un .
Thus we extend
(4.2)
∞
X
1
L(x1 , . . . , xn ) = 1 +
Qm (u1 , . . . , un ).
m!
m=1
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We compute this explicitly. Fix m ≥ 2. Assume that u1 , . . . , un ≥ 0 are
all unequal. We claim that if 2 ≤ n ≤ m + 1, then Cm−n+1 (u1 , . . . , un ) is
the (n − 1)th divided difference of the function f (u) = um with arguments
u1 , . . . , un . In other words,
(4.3) Cm−n+1 (u1 , . . . , un ) =
Cm−n+2 (u2 , . . . , un ) − Cm−n+2 (u1 , . . . , un−1 )
.
un − u 1
(For n = 2, we have simply Cm−1 (u1 , u2 ) =
To prove this, note that for k ≥ 1
m
um
2 −u1
.)
u2 −u1
(4.4) Ck (u1 , . . . , un ) = ukn + uk−1
n C1 (u1 , . . . , un−1 )
+ · · · + un Ck−1 (u1 , . . . , un−1 ) + Ck (u1 , . . . , un−1 )
and
Extending Means of Two
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Jorma K. Merikoski
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Ck (u1 , . . . , un ) = Ck (u1 , un ) + Ck−1 (u1 , un )C1 (u2 , . . . , un−1 )
+ · · · + C1 (u1 , un )Ck−1 (u2 , . . . , un−1 ) + Ck (u2 , . . . , un−1 ).
Hence,
Cm−n+2 (u2 , . . . , un ) − Cm−n+2 (u1 , . . . , un−1 )
= Cm−n+2 (u2 , . . . , un ) − Cm−n+2 (u2 , . . . , un−1 , u1 )
= um−n+2
+ um−n+1
C1 (u2 , . . . , un−1 ) + · · · + Cm−n+2 (u2 , . . . , un−1 )
n
n
m−n+2
m−n+1
− u1
− u1
C1 (u2 , . . . , un−1 ) − · · · − Cm−n+2 (u2 , . . . , un−1 )
)C1 (u2 , . . . , un−1 ) + · · ·
− um−n+2
) + (um−n+1
− um−n+1
= (um−n+2
1
n
n
1
+ (un − u1 )Cm−n+1 (u2 , . . . , un−1 )
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h
= (un − u1 ) Cm−n+1 (u1 , un ) + Cm−n (u1 , un )C1 (u2 , . . . , un−1 ) + · · ·
i
+ Cm−n+1 (u2 , . . . , un−1 )
= (un − u1 )Cm−n+1 (u1 , . . . , un ),
and (4.3) follows.
By a well-known formula of divided differences (see e.g. [4, p. 148]), we
now have
n
X
um
i
,
Cm−n+1 (u1 , . . . , un ) =
U
i
i=1
where
n
Y
Ui =
(ui − uj ).
j=1
j6=i
Therefore, since
−1
n + (m − n + 1) − 1
(n − 1)!
1
=
,
(m − n + 1)!
m−n+1
m!
we obtain
(n − 1)!
1
Qm−n+1 (u1 , . . . , un ) =
Cm−n+1 (u1 , . . . , un )
(m − n + 1)!
m!
n
(n − 1)! X um
i
.
=
m!
U
i
i=1
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Hence, and because the mth divided difference of the function f (u) = um is 1
if m = n − 1 and 0 if m ≤ n − 2, we have
∞
X
1
L(x1 , . . . , xn ) = 1 +
Qk (u1 , . . . , un )
k!
k=1
=1+
∞
X
1
Qm−n+1 (u1 , . . . , un )
(m − n + 1)!
m=n
∞
n
X
1 X um
i
= 1 + (n − 1)!
m! i=1 Ui
m=n
∞
X
n
1 X um
i
= (n − 1)!
m!
U
i
m=n−1
i=1
n
∞
X
1 X um
i
= (n − 1)!
m!
U
i
m=0
i=1
∞
n
n
X
X
eui
1 X um
i
= (n − 1)!
= (n − 1)!
Ui m=0 m!
Ui
i=1
i=1
= (n − 1)!
= (n − 1)!
n
X
i=1
n
X
i=1
Thus (1.3) is found.
eui
Qn
j=1 (ui − uj )
j6=i
xi
Qn
.
j=1 (ln xi − ln xj )
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Jorma K. Merikoski
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j6=i
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5.
Numerical Computation of L
Mustonen [7] noted that, in computing L numerically, the explicit formula (1.3)
is very unstable. He programmed a fast and stable algorithm based on (4.1),
(4.2), and (4.4). His experiments lead to a conjecture that, denoting Gn =
G(1, . . . , n) and Ln = L(1, . . . , n), we have
lim (Gn+1 − Gn ) = lim (Ln+1 − Ln ) =
n→∞
n→∞
1
e
Extending Means of Two
Variables to Several Variables
and
Gn
Ln
1
= lim
= .
n→∞
n
n
e
For Gn , these limit conjectures can be proved by using Stirling’s formula. For
Ln , they remain open.
lim
Jorma K. Merikoski
n→∞
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6.
Inequality G ≤ L ≤ A
It is natural to ask, whether
(6.1)
G(x1 , . . . , xn ) ≤ L(x1 , . . . , xn ) ≤ A(x1 , . . . , xn )
is generally valid.
For n = 2, this inequality is old (see e.g. [1, pp. 168-169]). Carlson [2] (see
also [1, p. 388]) sharpened the first part and Lin [5] (see also [1, p. 389]) the
second:
(6.2)
1
(G(x1 , x2 )M1/2 (x1 , x2 )) 2 ≤ L(x1 , x2 ) ≤ M1/3 (x1 , x2 ).
Neuman [9] defined (as a special case of [9, Eq. (2.3)])
!
Z
n
X
(6.3)
L(x1 , . . . , xn ) =
exp
ui ln xi du,
En−1
i=1
where u1 + · · · + un = 1,
En−1 = {(u1 , . . . , un−1 ) | u1 , . . . , un−1 ≥ 0, u1 + · · · + un−1 ≤ 1},
and du = du1 · · · dun−1 . He ([9], Theorem 1 and the last formula) proved (6.1)
and reduced (6.3) into (1.3).
Pečarić and Šimić [12] tied Neuman’s approach to a wider context. As a
special case ([12, Remark 5.4]), they obtained (1.3).
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Let V denote the Vandermonde determinant and let Vi denote its subdeterminant obtained by omitting its last row and ith column. Xiao and Zhang [14]
(unaware of [9]) defined
n
X
(n − 1)!
L(x1 , . . . , xn ) =
(−1)n+i xi Vi (ln x1 , . . . , ln xn ),
V (ln x1 , . . . , ln xn ) i=1
which in fact equals to (1.3). Also they proved (6.1).
We conjecture that (6.2) can be extended to
Extending Means of Two
Variables to Several Variables
1
(G(x1 , . . . , xn )M1/2 (x1 , . . . , xn )) 2 ≤ L(x1 , . . . , xn ) ≤ M1/3 (x1 , . . . , xn ).
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7.
Inequalities Pm[G] ≤ Pm[L] ≤ Pm[A]
In view of (3.2) and (3.3), it is now natural to ask, whether (6.1) can be strengthened to hold termwise. In other words: Do we have
Pm [G] ≤ Pm [L] ≤ Pm [A]
or equivalently
Rm [G] ≤ Rm [L] ≤ Rm [A],
Extending Means of Two
Variables to Several Variables
that is
(7.1)
1
u1 + · · · + u n
≤ Qm (u1 , . . . , un ) m ≤
n
m
um
1 + · · · + un
n
m1
for all u1 , . . . , un ≥ 0, m ≥ 1?
1
Fix u1 , . . . , un and denote qm = Qm (u1 , . . . , un ) m . Neuman ([8, Corollary
3.2]; see also [1, pp. 342-343]) proved that
(7.2)
k ≤ m ⇒ qk ≤ qm .
Jorma K. Merikoski
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The first part of (7.1), q1 ≤ qm , is therefore true. We conjecture that the second
part is also true.
DeTemple and Robertson [3] gave an elementary proof of (7.2) for n = 2,
but Neuman’s proof for general n is advanced, applying B-splines.
Mustonen [7] gave an elementary proof of (7.1) for n = 2.
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8.
Other Means
Pečarić and Šimić [12] (see also [1, p. 393]) studied a very large class of means,
called Stolarsky-Tobey means, which includes all the ordinary means as special
cases. They first defined these means for two variables and then, applying certain integrals, extended them to n variables. It might be of interest to apply
our method to all these extensions, but we take only a small step towards this
direction.
Let r and s be unequal nonzero real numbers. (Actually [12] allows s = 0
and [1] allows r = 0, both of which are obviously incorrect.) Consider ([12,
Eq. (6)]) the mean
Er,s (x1 , x2 ) =
(8.1)
r xs1 − xs2
·
s xr1 − xr2
Extending Means of Two
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1
s−r
,
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where x1 6= x2 . Assuming that s 6= −r, −2r, . . . , −(n − 2)r, this can be
extended ([12, Theorem 5.2(i)]) to
(8.2)
Er,s (x1 , . . . , xn )
"
=
n−1
(n − 1)! r
s(s + r) · · · (s + (n − 2)r)
where all the xi ’s are unequal.
n
X
s+(n−2)r
xi
Qn
r
r
j=1 (x − x )
i
j
i=1
j6=i
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1
# s−r
,
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To extend (8.1) by our method, we simply note that
1
s−r
xs1 − xs2
xr1 − xr2
Er,s (x1 , x2 ) =
s(ln x1 − ln x2 ) r(ln x1 − ln x2 )
1
L(xs1 , xs2 ) s−r
=
,
L(xr1 , xr2 )
which can be immediately extended to
(8.3) Er,s (x1 , . . . , xn )
1
L(xs1 , . . . , xsn ) s−r
=
L(xr1 , . . . , xrn )
1
( n
, n
) s−r
s
r
X
X
xi
xi
Qn
Qn
=
j=1 [s(ln xi − ln xj )]
j=1 [r(ln xi − ln xj )]
i=1
i=1
j6=i
j6=i
1
"
,
# s−r
n
n
r n−1 X
X
xsi
xri
Qn
Qn
.
=
s
j=1 (ln xi − ln xj )
j=1 (ln xi − ln xj )
i=1
i=1
j6=i
j6=i
This is obviously different from (8.2).
Unfortunately the problem of whether (8.3) indeed is a mean, i.e., whether it
lies between the smallest and largest xi , remains open.
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Addendum
Neuman ([10, Theorem 6.2]) proved the second part of (7.1) and [11] showed
that (8.3) is a mean.
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References
[1] P.S. BULLEN, Handbook of Means and Their Inequalities, Kluwer, 2003.
[2] B.C. CARLSON, The logarithmic mean, Amer. Math. Monthly, 79 (1972),
615–618.
[3] D.W. DeTEMPLE AND J.M. ROBERTSON, On generalized symmetric
means of two variables, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.
Fiz. No. 634-677 (1979), 236-238.
[4] C.E. FRÖBERG, Introduction to Numerical Analysis, Addison-Wesley,
1965.
[5] T.P. LIN, The power mean and the logarithmic mean, Amer. Math.
Monthly, 81 (1974), 879–883.
[6] S. MUSTONEN, A generalized logarithmic mean, Unpublished
manuscript, University of Helsinki, Department of Statistics, 1976.
[7] S. MUSTONEN, Logarithmic mean for several arguments, (2002). ONLINE [http://www.survo.fi/papers/logmean.pdf].
[8] E. NEUMAN, Inequalities involving generalized symmetric means, J.
Math. Anal. Appl., 120 (1986), 315–320.
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[9] E. NEUMAN, The weighted logarithmic mean, J. Math. Anal. Appl., 188
(1994), 885–900.
[10] E. NEUMAN, On complete symmetric functions, SIAM J. Math. Anal., 19
(1988), 736–750.
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[11] E. NEUMAN, Private communication (2004).
[12] J. PEČARIĆ AND V. ŠIMIĆ, Stolarsky-Tobey mean in n variables, Math.
Ineq. Appl., 2 (1999), 325–341.
[13] A.O. PITTENGER, The logarithmic mean in n variables, Amer. Math.
Monthly, 92 (1985), 99–104.
[14] Z-G. XIAO AND Z-H. ZHANG, The inequalities G ≤ L ≤ I ≤ A in
n variables, J. Ineq. Pure Appl. Math., 4(2) (2003), Article 39. ONLINE
[http://jipam.vu.edu.au/article.php?sid=277].
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