Journal of Inequalities in Pure and
Applied Mathematics
MONOTONIC REFINEMENTS OF A KY FAN INEQUALITY
KWOK KEI CHONG
Department of Applied Mathematics,
The Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong
The People’s Republic of China.
EMail: makkchon@inet.polyu.edu.hk
volume 2, issue 2, article 19,
2001.
Received 03 October, 2000;
accepted 02 February, 2001.
Communicated by: F. Qi
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
It is well-known that inequalities between means play a very important role in
many branches of mathematics. Please refer to [1, 3, 7], etc. The main aims of
the present article are:
(i) to show that there are monotonic and continuous functions H(t), K(t), P (t)
and Q(t) on [0, 1] such that for all t ∈ [0, 1],
Hn ≤ H(t) ≤ Gn ≤ K(t) ≤ An and
Hn /(1 − Hn ) ≤ P (t) ≤ Gn /G0n ≤ Q(t) ≤ An /A0n ,
where An , Gn and Hn are respectively the weighted arithmetic, geometric and harmonic means of the positive numbers x1 , x2 , ..., xn in (0, 1/2],
with positive weights α1 , α2 , ..., αn ; while A0n and G0n are respectively the
weighted arithmetic and geometric means of the numbers 1 − x1 , 1 −
x2 , ..., 1 − xn with the same positive weights α1 , α2 , ..., αn ;
(ii) to present more general monotonic refinements for the Ky Fan inequality
as well as some inequalities involving means; and
(iii) to present some generalized and new inequalities in this connection.
2000 Mathematics Subject Classification: 26D15, 26A48.
Key words: Ky Fan inequality, monotonic refinements of inequalities, arithmetic, geometric and harmonic means.
The author would like to thank the referees for their invaluable comments and suggestions.
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2
Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3
Monotonic Refinements of the Ky Fan Inequality . . . . . . . . . . 15
4
Second Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 21
References
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
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1.
Introduction
Let n be a positive integer. To two given sequences of positive numbers x1 ,
x2 , . . . , xn and α1 , α2 , . . . , αn such that α1 + α2 + · · · + αn = 1, we denote by
An , Gn and Hn respectively the weighted arithmetic, geometric and harmonic
means, that is,
n
X
An =
αi xi ,
i=1
Gn =
n
Y
Monotonic Refinements of a Ky
Fan Inequality
xαi i ,
i=1
Hn =
n
X
K. K. Chong
!−1
αi /xi
.
i=1
We use the symbols an , gn and hn to denote the corresponding unweighted
arithmetic, geometric and harmonic means of the n positive numbers
x1 , x2 , . . . , xn . The following well-known inequality has been proved, using
many different methods: (Please refer to [3].)
H n ≤ G n ≤ An
(1.1)
Let the real numbers xi be such that 0 < xi ≤ 1/2, for all i = 1, 2, . . . , n.
We denote by A0n , G0n and Hn0 the weighted arithmetic, geometric and harmonic
means of the numbers 1 − x1 , 1 − x2 , . . . , 1 − xn , namely,
A0n
=
n
X
i=1
αi (1 − xi ) ,
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G0n =
n
Y
(1 − xi )αi ,
i=1
Hn0 =
n
X
!−1
αi / (1 − xi )
.
i=1
a0n ,
gn0
h0n
Also, let
and
denote the corresponding unweighted arithmetic, geometric and harmonic means of the numbers 1 − x1 , 1 − x2 , . . . , 1 − xn respectively. In recent years many interesting inequalities involving these mean
values have been published, in particular, the following well-known Ky Fan and
Wang-Wang inequalities :
(1.2)
Gn
Hn
An
0 ≤
0 ≤
Hn
Gn
A0n
with equality holding if and only if x1 = · · · = xn . Please refer to the following
papers by H. Alzer [1] – [2] and Wang-Wang [10] or [7], etc. The right-hand
inequality of (1.2) is the famous Ky Fan inequality; the left-hand inequality for
the unweighted case was first discovered by Wang-Wang in 1984 [10]. The main
purpose of this paper is to present some new monotonic continuous functions
H(λ), K(λ), P (λ) and Q(λ) on [0, 1] such that
Hn ≤ H(λ) ≤ Gn ≤ K(λ) ≤ An
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
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and
Hn /(1 − Hn ) ≤ P (λ) ≤ Gn /G0n ≤ Q(λ) ≤ An /A0n .
In fact, our theorems here generalize results of Wang and Yang in [9] and
a theorem of K.M. Chong [6]. In Section 2, we shall generalize refinements
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of inequalities between means. In Section 3, we shall present generalizations
to refinements of the Ky Fan inequality, with some new inequalities deduced.
Finally, in Section 4, we shall show that Theorem 3.1 can be used to deduce
many other established refinements of the Ky Fan inequality.
In a recent paper of Wang and Yang [9], the following two interesting theorems were put forward. In fact, they are refinements of inequalities (1.1) and
(1.2) in Section 1, for the discrete unweighted case. They are restated here
without proof. For the details of the proof, please refer to [9].
Theorem 1.1. Given a sequence { x1 , x2 , . . . , xn } of positive numbers, which
are not all equal:
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
(a) For any t in [0, 1/n] , let
(1.3)
h (t) =
n
Y
i=1
"
n X
1
1
1
+t
−
xi
x
xi
j
j=1
#−1/n
Then, h(t) is continuous, strictly decreasing and hn = h(1/n) ≤ h(t) ≤
h(0) = gn on [0, 1/n] .
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(b) For any t in [0, 1/n] , let
Close
(1.4)
k (t) =
n
Y
i=1
"
xi + t
n
X
#1/n
(xj − xi )
j=1
Then, k(t) is continuous, strictly increasing and gn = k(0) ≤ k(t) ≤
k(1/n) = an on [0, 1/n] .
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Theorem 1.2. Given a sequence { x1 , x2 , . . . , xn } with xi in (0, 1/2], i =
1, 2, . . . , n, which are not all equal:
(a) For any t in [0, 1/n] , let
(1.5)
p (t) =
n
Y
i=1
"
#−1/n
n X
1
1
1
+t
−
−1
xi
xj
xi
j=1
Then, p(t) is continuous, strictly decreasing, and hn /(1 − hn ) = p(1/n)
≤ p(t) ≤ p(0) = gn /gn0 on [0, 1/n] .
K. K. Chong
(b) For any t in [0, 1/n] , let
n
Q
(1.6)
q (t) =
Monotonic Refinements of a Ky
Fan Inequality
i=1
n
Q
i=1
"
xi + t
n
P
#1/n
(xj − xi )
Contents
j=1
"
1 − xi − t
n
P
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#1/n
(xj − xi )
j=1
Then, q(t) is continuous, strictly increasing and gn /gn0 = q(0) ≤ q(t) ≤
q(1/n) = an /a0n on [0, 1/n] .
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2.
Some Generalizations
In this section, we are going to present and prove a generalization of Theorem 1.1 and Theorem 1.2(a), in particular, to the case for weighted means. Its
statement runs as follows:
Theorem 2.1. Let a1 , a2 , . . . , an , and α1 , P
α2 , . . . , αn be two sequences of positive numbers, with ai not all equal and P ni=1 αi = 1. Let a be any positive
number such that An ≤ a, where An = ni=1 αi ai , and k is a constant such
that k < ai , for all i = 1, 2, . . . , n. Let
(2.1)
F (λ) =
n
Y
[λa + (1 − λ) ai − k]αi
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
i=1
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(2.2)
G (λ) =
n
Y
[λa + (1 − λ) ai − k]−αi
i=1
Then,
(i) F (λ) is continuous and strictly increasing on [0, 1] ;
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(ii) G(λ) is continuous and strictly decreasing on [0, 1] .
Proof.
(i) Taking the logarithm of F (λ), we have,
ln F (λ) =
n
X
i=1
αi ln [λa + (1 − λ) ai − k]
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Differentiating the last expression with respect to λ, we have:
n
F 0 (λ) X
αi (a − ai )
=
F (λ)
[λa + (1 − λ) ai − k]
i=1
Differentiating again, we obtain:
(2.3)
F 0 (λ)
F (λ)
0
=−
n
X
i=1
αi (a − ai )2
<0
[λa + (1 − λ) ai − k]2
for all λ in [0, 1] , as the ai are not all equal. Hence, F 0 (λ)/F (λ) is strictly
decreasing on [0, 1] . Also, as An ≤ a and k < a, we have :
n
(2.4)
a − An
F 0 (1) X αi (a − ai )
=
=
≥0
F (1)
a
−
k
a
−
k
i=1
Therefore, F 0 (λ)/F (λ) > 0, for all λ in [0, 1). As F (λ) is positive for all
λ in [0, 1] , F 0 (λ) > 0 for λ in [0, 1). Hence, F (λ) is strictly increasing on
[0, 1] . The continuity of F (λ) on [0, 1] is obvious.
(ii) As F (λ) is positive for all λ in [0, 1] and G(λ) = 1/F (λ), G(λ) is continuous and strictly decreasing on [0, 1]. Hence, the proof of Theorem 2.1
is complete.
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
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Now, we use Theorem 2.1 to deduce some established theorems.
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Remark 2.1.
(i) From Theorem 2.1, we have, for all λ ∈ (0, 1),
F (0) < F (λ) < F (1),
which yields for not all equal ai ,
(2.5)
n
Y
(ai − k)αi < a − k.
i=1
In particular, if a = An , for not all equal ai we have,
(2.6)
n
Y
(ai − k)αi < An − k,
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
i=1
which is a generalization of the weighted arithmetic-geometric means inequality.
P
(ii) Again, in Theorem 2.1, we let k = 0, a = ni=1 αi ai = An . Then, F (λ)
will reduce to, say
(2.7)
K(λ) =
n
Y
[λAn + (1 − λ)ai ]αi
i=1
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It is clear that K(λ) is continuous and strictly increasing on [0, 1] , and for
all λ ∈ (0, 1) ,
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K(0) = Gn < K(λ) < K(1) = An .
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(2.8)
This is a refinement of the weighted arithmetic-geometric means inequality.
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(iii) Furthermore, if we put λ = nt, αi =
obtain for all t ∈ [0, 1/n] ,
K(nt) =
n
Y
i=1
1
,i
n
[ntAn + (1 − nt)ai ]1/n =
= 1, 2, . . . , n, into K(λ), we
n
Y
"
ai + t
i=1
n
X
#1/n
(aj − ai )
.
j=1
The last expression is in fact the function k(t) of Theorem 1.1(b). Hence,
we have shown that Theorem 1.1(b) is a particular case of Theorem 2.1.
Remark 2.2. If, in Theorem 2.1, we let k = 0, ai = x1i , i = 1, 2, . . . , n, a =
1
= αx11 + · · · + αxnn , then G(λ) will reduce to,
Hn
−αi
n Y
1
λ
(2.9)
H(λ) =
+ (1 − λ)
.
H
x
n
i
i=1
Then, H(λ) is continuous and strictly decreasing on [0, 1] , and for all λ ∈
(0, 1),
(2.10)
H(1) = Hn < H(λ) < H(0) = Gn .
(2.10) is a refinement of the weighted means inequality. Furthermore, if we put
λ = nt, αi = n1 , ai = 1/xi , i = 1, 2, . . . , n, into H(λ), we obtain for all t in
[0, 1/n] ,
"
−1/n Y
#−1/n
n n
n Y
X
1
1
1
1
H (nt) =
nta + (1 − nt)
=
+t
−
x
x
x
xi
i
i
j
i=1
i=1
j=1
This is the function h(t) in Theorem 1.1(a). Hence, we have deduced Theorem
1.1(a) as a particular case of Theorem 2.1.
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
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Theorem 2.2. Let x1 , x2 , . . . , xn be n positive numbers, not all equal, with
xi ∈ (0, 1/2] for all i = 1, 2, .., n. Let α1 , α2 , . . . , αn be the corresponding
weights, i.e., αi > 0, i = 1, 2, . . . , n and α1 + · · · + αn = 1. Let γ be a constant
such that γ < x1i , for all i = 1, 2, . . . , n. We define P (λ) as :
−αi
n Y
λ
1
+ (1 − λ) − γ
(2.11)
P (λ) =
Hn
xi
i=1
Then,
Monotonic Refinements of a Ky
Fan Inequality
(i) P (λ) is continuous and strictly decreasing on [0, 1] ;
(ii) for all λ ∈ (0, 1), we have,
K. K. Chong
n
(2.12)
Y
Hn
P (1) =
< P (λ) < P (0) =
1 − γHn
i=1
xi
1 − γxi
α i
.
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Proof. (i) P (λ) is continuous and strictly decreasing on [0, 1] , as we get P (λ)
from the continuous and strictly decreasing function G(λ), by putting k =
γ, ai = 1/xi with xi ∈ (0, 1/2], i = 1, 2, . . . , n, a = 1/Hn = α1 /x1 +
· · · + αn /xn into G(λ) of Theorem 2.1.
Q xi α i
(ii) We have : P (0) = G(0) = ni=1 1−γx
,
i
P (1) = G(1) = Hn /(1 − γHn ).
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Hence, for all λ ∈ (0, 1),
Hn /(1 − γHn ) < P (λ) <
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n Y
i=1
xi
1 − γxi
α i
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This completes the proof of Theorem 2.2.
Remark 2.3. If we put λ = nt, αi = 1/n, i = 1, 2, . . . , n, γ = 1 into P (λ) of
Theorem 2.2, we obtain for any t in [0, 1/n] ,
−1/n
n Y
nt
P (nt) =
+ (1 − nt) ai − 1
hn
i=1
"
#−1/n
n
n Y
X
1
1
1
=
+t
−
−1
x
x
x
i
j
i
i=1
j=1
Monotonic Refinements of a Ky
Fan Inequality
This is the function p(t) of Theorem 1.2(a), and we have deduced Theorem
1.2(a) as a particular case of Theorem 2.1.
K. K. Chong
The only part in Section 1, which is not yet dealt with, is Theorem 1.2(b).
Its proof is postponed to the next section, with some additional theorems. We
end this section by considering another similar theorem. In [6], K.M. Chong
presented the following theorem:
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Theorem 2.3. Let a1 , a2 , . . . , an be positive numbers and α1 , P
α2 , . . . , αn be
their corresponding weights, i.e. αi > 0, i = 1, 2, . . . , n and ni=1 αi = 1.
Let f (λ) be defined as:
" n
#αi
n
Y
X
(2.13)
f (λ) =
λ
αj aj + (1 − λ) ai
i=1
j=1
Then f (λ) is a strictly increasing function of λ for λ ∈ [0, 1], unless a1 = a2 =
· · · = an ; in which case f (0) = Gn = An = f (1) .
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Proof. It is obvious that when k = 0 and a = An in Theorem 2.1 we obtain
K.M. Chong’s theorem at once.
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
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3.
Monotonic Refinements of the Ky Fan Inequality
In the previous section, we have seen that Theorem 2.1 is a generalization of
various theorems. In this section, we shall present a refinement of the wellknown Ky Fan inequality, which is a generalization of Theorem 1.2(b).
Theorem 3.1. Let x1 , x2 , . . . , xn be n positive numbers, not all equal, such that
xi ∈ (0, 1/2] for all i = 1, 2, . . . , n and let αP
1 , α2 , . . . , αn be their corresponding weights i.e. αi > 0, i = 1, 2, . . . , n and ni=1 αi = 1. Let β and δ be two
constants such that δ ≤ β and β < xi , for all i = 1, 2, . . . , n. Let r(λ) be
defined as :
n
Q
r(λ) = Q
n
(3.1)
i=1
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[λ(1 − z) + (1 − λ)(1 − xi ) − δ]αi
i=1
Pn
i=1
αi xi ≤ z ≤ 1/2.
(i) r(λ) is continuous and strictly increasing on [0, 1] ; and
(ii) when β = δ = 0, we have, for all λ ∈ (0, 1) ,
(3.2)
K. K. Chong
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[λz + (1 − λ)xi − β]αi
for any λ ∈ [0, 1], and any real number z such that
Then,
Monotonic Refinements of a Ky
Fan Inequality
z
Gn /Gn = r(0) < r(λ) < r(1) =
.
1−z
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Proof.
(i) Taking the logarithm, we have,
ln{r(λ)} =
n
X
αi ln[λz + (1 − λ)xi − β]
i=1
−
n
X
αi ln[λ(1 − z) + (1 − λ)(1 − xi ) − δ]
i=1
Differentiating with respect to λ, we have :
r0 (λ)
r(λ)
=
K. K. Chong
n
X
i=1
=
Monotonic Refinements of a Ky
Fan Inequality
n
X
i=1
n
X
z − xi
(1 − z) − (1 − xi )
−
αi
αi
λz + (1 − λ)xi − β
λ(1 − z) + (1 − λ)(1 − xi ) − δ
i=1
n
X
z − xi
z − xi
αi
+
αi
.
λz + (1 − λ)xi − β
λ(1 − z) + (1 − λ)(1 − xi ) − δ
i=1
Let u(λ) =
r 0 (λ)
.
r(λ)
We are going to show that ln{r(λ)} and hence r(λ) are both strictly increasing by showing that u(λ) > 0 for all λ ∈ [0, 1).
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Differentiating u(λ) with respect to λ, we have :
0
u (λ) = −
n
X
i=1
αi (z − xi )2
[λz + (1 − λ)xi − β]2
+
n
X
i=1
αi (z − xi )2
<0
[λ(1 − z) + (1 − λ)(1 − xi ) − δ]2
as
1
1
>
2
[λz + (1 − λ)xi − β]
[λ(1 − z) + (1 − λ)(1 − xi ) − δ]2
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
i = 1, 2, . . . , n, unless z = x1 = x2 = . . . = xn = 1/2, and β = δ.
Hence u(λ) is strictly decreasing on [0, 1] .
u(1) =
n
X
i=1
n
z − xi
z − xi X
αi
+
αi
z−β
1−z−δ
i=1
n
n
X
1 X
1
αi (z − xi ) +
αi (z − xi )
z − β i=1
1 − z − δ i=1
!
n
n
X
X
1−β−δ
≥ 0, for
αi xi ≤ z ≤ 1/2.
= z−
αi xi
(z
−
β)(1
−
z
−
δ)
i=1
i=1
=
Hence, u(λ) =
r 0 (λ)
r(λ)
> 0, for all λ ∈ [0, 1).
As r(λ) is always positive, we have r0 (λ) > 0 for all λ ∈ [0, 1) and r(λ)
is strictly increasing on [0, 1] .
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0
(ii) It is easy to see that when β = δ = 0, r(0) = Gn /Gn , r(1) =
r(0) < r(λ) < r(1) on (0, 1).
z
1−z
and
P
It is remarked, that if β = δ = 0, z = ni=1 αi xi in Theorem 3.1, then the
chain of inequalities in (3.2), with r(λ) replaced by Q(λ), will become : for any
λ ∈ (0, 1),
(3.3)
z
Gn
An
= 0
0 = Q(0) < Q(λ) < Q(1) =
Gn
1−z
An
This is a refinement of the Ky Fan inequality.
Remark 3.1. (3.3) is a refinement of the weighted Ky Fan inequality and we
have r(0) < r(λ) < r(1), unless x1 = x2 = . . . = xn . In general, (3.2) yields a
generalization of the Ky Fan inequality as follows :
For An ≤ z ≤ 1/2 and δ ≤ β < xi ∈ (0, 1/2], for all i = 1, 2, . . . , n, we
have,
n
Q
(3.4)
[xi − β]αi
z−β
i=1
≤
= r(1),
r(0) = Q
n
1−z−δ
[1 − xi − δ]αi
i=1
with equality if and only if x1 = x2 = · · · = xn .
If in (3.4) we let z = An the weighted arithmetic mean of x1 , x2 , . . . , xn , we
Monotonic Refinements of a Ky
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K. K. Chong
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obtain a generalization of the weighted Ky Fan inequality :
n
Q
[xi − β]αi
i=1
(3.5)
n
Q
≤
[1 − xi − δ]αi
An − β
,
A0n − δ
i=1
with equality if and only if x1 = x2 = · · · = xn .
P
Remark 3.2. If we put αi = 1/n, i = 1, 2, . . . , n, z = ni=1 αi xi , β = δ = 0
and λ = nt into (3.1), we then obtain after simplification,
"
#1/n
n
n 1
Q
P
nt
xj + (1 − nt)xi
i=1
j=1 n
r(nt) =
"
!
#1/n
n
n 1
Q
P
nt 1 −
xj + (1 − nt)(1 − xi )
i=1
j=1 n
"
#1/n
n
n
Q
P
xi + t (xj − xi )
=
i=1
n
Q
i=1
j=1
"
1 − xi − t
n
P
#1/n = q(t),
for t ∈ [0, 1/n].
(xj − xi )
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K. K. Chong
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This is the function q(t) in Theorem 1.2(b), showing that Theorem 3.1 is a generalization of Theorem 1.2(b).
Remark 3.3. In [5], we have the following theorem, which can be easily seen
to follow as a particular case of Theorem 3.1, when β = 0 and δ = 0 :
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Theorem 3.2. Let x1 , x2 , . . . , xn be n positive numbers, such that xi ∈ (0, 1/2],
for all i = 1, 2, . . . , n, and let α1 , α2 , . . . , αn be their corresponding positive
weights, with α1 + α2 P
+ · · · + αn = 1. Let z be a constant such that An ≤
z ≤ 1/2, where An = ni=1 αi xi . We define w(λ), for any λ ∈ [0, 1], to be the
function:
n
Q
w(λ) = Q
n
(3.6)
[λz + (1 − λ)xi ]αi
i=1
.
αi
[λ(1 − z) + (1 − λ)(1 − xi )]
i=1
Monotonic Refinements of a Ky
Fan Inequality
K. K. Chong
Then,
(i) w(λ) is continuous and strictly increasing on [0, 1] , unless x1 = x2 =
· · · = xn ;
0
(ii) Gn /Gn = w(0) ≤ w(λ) ≤ w(1) =
z
,
1−z
for λ ∈ [0, 1] .
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4.
Second Proof of Theorem 2.1
In this section, we shall show that Theorem 3.1 is not only a generalization of
Theorem 1.2(b), but also it can be used to deduce some elementary theorems.
Second proof of Theorem 2.1. Suppose the n numbers x1 , x2 , . . . , xn are not all
equal.
P
For xi ∈ (0, 1/2], i = 1, 2, . . . , n, z lying between ni=1 αi xi and 1/2, the
function r(λ) of Theorem 3.1 is strictly increasing on [0, 1] , where for λ ∈
[0, 1] , r(λ) is defined as :
n
Q
[λz + (1 − λ)xi − β]αi
i=1
(4.1)
r(λ) = Q
.
n
[λ(1 − z) + (1 − λ)(1 − xi ) − δ]αi
i=1
Now, we put xi = ali , i = 1, 2, . . . , n, where l is a large positive number, and
let z = al , β = kl with δ = β. Then, the function r(λ) becomes :
(4.2)
n
1Q
[λa + (1 − λ)ai − k]αi
l i=1
r(λ) = Q
iα i .
n h
a
ai
λ(1 − ) + (1 − λ)(1 − ) − β
l
l
i=1
By Theorem 3.1,
v(λ) =
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n
Q
(4.3)
Monotonic Refinements of a Ky
Fan Inequality
[λa + (1 − λ)ai − k]αi
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i=1
α
a
ai
k i
λ(1 − ) + (1 − λ)(1 − ) −
l
l
l
i=1
n
Q
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is strictly increasing as λ increases from 0 to 1.
We let l tend to +∞, the denominator tends to 1 and we have shown that the
function in Theorem 2.1
(4.4)
n
Y
F (λ) =
[λa + (1 − λ)ai − k]αi
i=1
is an increasing function on [0, 1] .
Differentiation calculations as in Theorem 2.1 easily reveal that in fact F (λ)
is strictly increasing on [0, 1] . This completes the proof of Theorem 2.1.
Remark 4.1. From the discussions in Section 2, it can be seen that Theorem
2.1 generalizes Theorem 1.1, Theorem 1.2(a), Theorem 2.2 and Theorem 2.3.
From the discussions of the last two sections, it can be seen that Theorem 3.1
generalizes Theorem 1.2(b), Theorem 3.2 and Theorem 2.1. As a whole, we
have shown that Theorem 3.1 is a generalization of all other refinements of
inequalities (1.1) and (1.2), appearing in this paper.
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K. K. Chong
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References
[1] H. ALZER, Inequalities for arithmetic, geometric and harmonic means,
Bull. London Math. Soc., 22 (1990), 362–366.
[2] H. ALZER, The inequality of Ky Fan and related results, Acta Appl. Math.,
38 (1995), 305–354.
[3] P.S. BULLEN, D.S. MITRINOVIĆ AND J.E. PEČARIĆ, Means and Their
Inequalities, Reiddel Dordrecht, 1988.
[4] K.K. CHONG, On a Ky Fan’s inequality and some related inequalities
between means, Southeast Asian Bull. Math., 22 (1998), 363–372.
[5] K.K. CHONG, On some generalizations and refinements of a Ky Fan inequality, Southeast Asian Bull. Math., 24 (2000), 355–364.
[6] K.M. CHONG, On the arithmetic-mean-geometric-mean inequality, Nanta
Mathematica, 10(1) (1977), 26–27.
[7] A.M. FINK, J.E. PEČARIĆ AND D.S. MITRINOVIĆ, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, 1993.
[8] N. LEVINSON, Generalization of an inequality of Ky Fan, J. Math. Anal.
Appl., 3 (1964), 133–134.
[9] C.S. WANG AND G.S. YANG, Refinements on an inequality of Ky Fan, J.
Math. Anal. Appl., 201 (1996), 955–965.
[10] W.L. WANG AND P.F. WANG, A class of inequalities for the symmetric
functions, Acta Math. Sinica, 27 (1984), 485–497 [In Chinese].
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K. K. Chong
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[11] W.L. WANG, Some inequalities involving means and their converses, J.
Math. Anal. Appl., 238 (1999), 567–579.
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K. K. Chong
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