Journal of Inequalities in Pure and
Applied Mathematics
GENERALIZATIONS OF THE KY FAN INEQUALITY
AI-JUN LI, XUE-MIN WANG AND CHAO-PING CHEN
Jiaozuo University
Jiaozuo City, Henan Province
454000, China
EMail: liaijun72@163.com
EMail: wangxm881@163.com
College of Mathematics and Informatics
Research Institute of Applied Mathematics
Henan Polytechnic University
Jiaozuo City, Henan 454010, China
EMail: chenchaoping@hpu.edu.cn
volume 7, issue 4, article 130,
2006.
Received 25 November, 2005;
accepted 06 October, 2006.
Communicated by: D. Ştefǎnescu
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
346-05
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Abstract
In this paper, we extend the Ky Fan inequality to several general integral forms,
Ls (a,b)
with α, a, b ∈
and obtain the monotonic properties of the function Ls (α−a,α−b)
(0, +∞) and s ∈ R.
2000 Mathematics Subject Classification: 26A48, 26D20.
Key words: Generalized logarithmic mean, Monotonicity, Ky Fan inequality.
Generalizations of the Ky Fan
Inequality
The authors were supported in part by the Science Foundation of the Project for
Fostering Innovation Talents at Universities of Henan Province, China
Ai-Jun Li, Xue-Min Wang and
Chao-Ping Chen
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Proofs of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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3
7
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1.
Introduction
The following inequality proposed by Ky Fan was recorded in [1, p. 5] : If
0 < xi ≤ 21 for i = 1, 2, . . . , n, then
(1.1)
Qn
xi
Qn
i=1 (1 − xi )
i=1
n1
Pn
xi
,
i=1 (1 − xi )
i=1
≤ Pn
unless x1 = x2 = · · · = xn .
With the notation
(1.2)
Mr (x) =
Generalizations of the Ky Fan
Inequality


1
n
Pn
i=1
xri
r1
(Qn x ) n1 ,
i=1 i
,
r 6= 0;
r = 0,
where Mr (x) denotes the r-order power mean of xi > 0 for i = 1, 2, . . . , n, the
inequality (1.1) can be written as
(1.3)
Ai-Jun Li, Xue-Min Wang and
Chao-Ping Chen
M0 (x)
M1 (x)
≤
.
M0 (1 − x)
M1 (1 − x)
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In 1996, Zh. Wang, J. Chen and X. Li [12] found the necessary and sufficient
condition for
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Mr (x)
Ms (x)
≤
Mr (1 − x)
Ms (1 − x)
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(1.4)
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Lr (a,b)
when r < s. Recently, Ch.-P. Chen proved that the function Lr (1−a,1−b)
is
1
1
strictly increasing for 0 < a < b ≤ 2 and strictly decreasing for 2 ≤ a < b < 1,
J. Ineq. Pure and Appl. Math. 7(4) Art. 130, 2006
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where r ∈ (−∞, ∞) and Lr (a, b) is the generalized logarithmic mean of two
positive numbers a, b, which is a special case of the extended means E(r, s; x, y)
defined by Stolarsky [10] in 1975. For more information about the extended
means please refer to [4, 6, 8, 11] and references therein.
Moreover, we have,
Lr (a, b) = a,
a = b;
r+1
r1
b
− ar+1
Lr (a, b) =
,
(r + 1)(b − a)
b−a
L−1 (a, b) =
= L(a, b);
ln b − ln a
1
1 bb b−a
L0 (a, b) =
= I(a, b),
e aa
a 6= b, r 6= −1, 0;
where L(a, b) and I(a, b) are respectively the logarithmic mean and the exponential mean of two positive numbers a and b. When a 6= b, Lr (a, b) is a strictly
increasing function of r. In particular,
lim Lr (a, b) = min{a, b},
r→−∞
L1 (a, b) = A(a, b),
lim Lr (a, b) = max{a, b},
r→+∞
L−2 (a, b) = G(a, b),
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where A(a, b) and G(a, b) are the arithmetic and the geometric means, respectively. For a 6= b, the following well known inequality holds:
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G(a, b) < L(a, b) < I(a, b) < A(a, b).
J. Ineq. Pure and Appl. Math. 7(4) Art. 130, 2006
(1.5)
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In this paper, motivated by inequality (1.4), we will extend the inequality
(1.4) to general integral forms. Some monotonic properties of several related
functions will be obtained.
Theorem 1.1. Let
Rb
fα (s) =
a
Rb
a
xs d x
(α − x)s d x
! 1s
=
Ls (a, b)
,
Ls (α − a, α − b)
s ∈ (−∞, +∞) and α be a positive number. Then fα (s) is a strictly increasing
function for [a, b] ⊆ (0, α2 ], and is a strictly decreasing function for [a, b] ⊆
[ α2 , α).
Generalizations of the Ky Fan
Inequality
Ai-Jun Li, Xue-Min Wang and
Chao-Ping Chen
Corollary 1.2. If [a, b] ⊆ (0, α2 ] and α is a positive number, then
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(1.6)
a
G(a, b)
L(a, b)
<
<
α−b
G(α − a, α − b)
L(α − a, α − b)
I(a, b)
A(a, b)
b
<
<
<
.
I(α − a, α − b)
A(α − a, α − b)
α−a
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If [a, b] ⊆ [ α2 , α), the inequalities (1.6) is reversed.
R
1s
b s
x dx
a
Corollary 1.3. Let hα (s) = R α−a xs d x , s ∈ (−∞, +∞) and α be a positive
α−b
number. Then hα (s) is a strictly increasing function for [a, b] ⊆ (0,
strictly decreasing function for [a, b] ⊆ [ α2 , α).
α
],
2
or a
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In [13], Feng Qi has proved that the function
r 7→
1
b−a
1
b+δ−a
Rb
a
xr d x
R b+δ
a
! r1
=
xr d x
Lr (a, b)
Lr (a, b + δ)
is strictly decreasing with r ∈ (−∞, +∞). Now, we will extend the conclusion
in the following theorem.
Theorem 1.4. Let
Generalizations of the Ky Fan
Inequality
f (s) =
1
b−a
1
d−c
Rb
a
Rd
c
xs d x
! 1s
=
xs d x
Ls (a, b)
,
Ls (c, d)
s ∈ (−∞, +∞) and a, b, c, d be positive numbers. Then f (s) is a strictly increasing function for ad < bc, or a strictly decreasing function for ad > bc.
Corollary 1.5. Let
h(s) =
1
b−a
1
d−a
Rb
a
Rd
a
xs d x
xs d x
! 1s
=
Ls (a, b)
,
Ls (a, d)
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Chao-Ping Chen
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s ∈ (−∞, +∞) and a, b, d are positive numbers. Then h(s) is a strictly increasing function for d < b, or a strictly decreasing function for d > b.
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2.
Proofs of Theorems
In order to prove Theorem 1.1, we make use of the following elementary lemma
which can be found in [3, p. 395].
Lemma 2.1 ([3, p. 395]). Let the second derivative of φ(x) be continuous with
x ∈ (−∞, ∞) and φ(0) = 0. Define

 φ(x) , x 6= 0;
x
(2.1)
g(x) =
 0
φ (0), x = 0.
Then φ(x) is strictly convex (concave) if and only if g(x) is strictly increasing
(decreasing) with x ∈ (−∞, ∞).
Remark 1. A general conclusion was given in [7, p. 18]: A function φ is convex
0)
is nondecreasing on [a, b] for every point x0 ∈
on [a, b] if and only if φ(x)−φ(x
x−x0
[a, b].
Proof of Theorem 1.1. It is obvious that
Rb
fα (s) =
a
Rb
a
xs d x
(α − x)s d x
! 1s
=
=
bs+1 − as+1
(α − a)s+1 − (α − b)s+1
Ls (a, b)
.
Ls (α − a, α − b)
1s
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Inequality
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Chao-Ping Chen
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Define for s ∈ (−∞, ∞),
 s+1
s+1
b
−
a


ln
, s 6= −1;

(α − a)s+1 − (α − b)s+1
(2.2)
ϕ(s) =

ln(b/a)

ln
,
s = −1.

ln[(α − a)/(α − b)]
Then
(2.3)

 ϕ(s) , s 6= 0;
s
ln fα (s) =
 0
ϕ (0), s = 0.
In order to prove that ln fα is strictly increasing (decreasing), it suffices to
show that ϕ is strictly convex (concave) on (−∞, ∞). A simple calculation
reveals that
(2.4)
ϕ(−1 − s) = ϕ(−1 + s) + s ln
(α − a)(α − b)
,
ab
which implies that ϕ00 (−1 − s) = ϕ00 (−1 + s), and ϕ has the same convexity
(concavity) on both (−∞, −1) and (−1, ∞). Hence, it is sufficient to prove that
ϕ is strictly convex (concave) on (−1, ∞).
A computation yields
ϕ0 (s) =
bs+1 ln b − as+1 ln a (α − b)s+1 ln(α − b) − (α − a)s+1 ln(α − a)
−
,
bs+1 − as+1
(α − b)s+1 − (α − a)s+1
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(2.5) (s + 1)2 ϕ00 (s)
"
= (s + 1)2
α−b 2
)
as+1 bs+1 (ln ab )2 (α − a)s+1 (α − b)s+1 (ln α−a
+
− s+1
s+1
2
s+1
s+1
2
(b − a )
[(α − a) − (α − b) ]
#
α−b s+1
α−b s+1 2
) [ln( α−a
) ]
( ab )s+1 [ln( ab )s+1 ]2 ( α−a
=−
.
+
α−b s+1 2
[1 − ( ab )s+1 ]2
[1 − ( α−a
) ]
Generalizations of the Ky Fan
Inequality
Define for 0 < t < 1,
ω(t) =
(2.6)
t(ln t)2
.
(1 − t)2
Ai-Jun Li, Xue-Min Wang and
Chao-Ping Chen
Differentiation yields
ω 0 (t)
(1 − t)t ln t
= (1 + t) ln t + 2(1 − t)
ω(t)
∞
X
n−1
=−
(1 − t)n+1 < 0,
n(n + 1)
n=2
(2.7)
which implies that ω 0 (t) > 0 for 0 < t < 1. It is easy to see that
a s+1 α − b s+1
αi
(2.8) 0 <
<
< 1 for [a, b] ⊆ 0,
, s > −1,
b
α−a
2
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(2.9)
0<
α−b
α−a
s+1
<
a s+1
b
< 1 for
[a, b] ⊆
hα
2
, α , s > −1,
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and therefore ϕ00 (s) > 0 for [a, b] ⊆ (0, α2 ] and s > −1, ϕ00 (s) < 0 for [a, b] ⊆
[ α2 , α) and s > −1. Then ϕ is strictly convex (concave) on (−1, ∞) for [a, b] ⊆
(0, α2 ] ([a, b] ⊆ [ α2 , α)) respectively. By Lemma 2.1 above, Theorem 1.1 holds.
Since fα (s) is a strictly increasing (decreasing) function for [a, b] ⊆ (0, α2 ]
([a, b] ⊆ [ α2 , α)), put s = −2, −1, 0, 1 respectively. The inequalities (1.6) are
deduced.
1s
Rb s
a x dx
Then, let (α − x) = t and apply it to the function R b (α−x)s d x . We get
a
Corollary 1.3.
Proof of Theorem 1.4. Using an analogous method of proof to that of Theorem
1.1, we get
f (s) =
1
b−a
1
d−c
Rb
a
Rd
c
s
x dx
xs d x
(s+1)(b−a)
ds+1 −cs+1
(s+1)(d−c)
=
(d − c) (bs+1 − as+1 )
=
(b − a) (ds+1 − cs+1 )
Let M =
(2.10)
(d−c)
,
(b−a)
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Chao-Ping Chen
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" bs+1 −as+1 # 1s
! 1s
Generalizations of the Ky Fan
Inequality
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1s
Ls (a, b)
=
.
Ls (c, d)
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and define for s ∈ (−∞, ∞),
 bs+1 − as+1



ln M ds+1 − cs+1 , s 6= −1;
ϕ(s) =

ln(b/a)


,
s = −1.
ln M
ln(d/c)
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Then

 ϕ(s) , s 6= 0;
s
ln f (s) =
 0
ϕ (0), s = 0,
(2.11)
and ϕ has the same convexity (concavity) on both (−∞, −1) and (−1, ∞).
A computation yields
( ab )s+1 [ln( ab )s+1 ]2 ( dc )s+1 [ln( dc )s+1 ]2
.
+
(s + 1) ϕ (s) = −
[1 − ( ab )s+1 ]2
[1 − ( dc )s+1 ]2
2
00
Define for 0 < t < 1,
ω(t) =
(2.12)
t(ln t)2
.
(1 − t)2
Differentiation yields ω 0 (t) > 0 for 0 < t < 1. It is easy to see that
a s+1 c s+1
(2.13)
0<
<
< 1 for ad < bc, s > −1,
b
d
c s+1
a s+1
< 1 for ad > bc, s > −1,
d
b
and therefore ϕ00 (s) > 0 for ad < bc and s > −1, ϕ00 (s) < 0 for ad > bc and
s > −1 Then ϕ is strictly convex (concave) on (−1, ∞) for ad < bc (ad > bc)
respectively. The proof is complete.
(2.14)
0<
<
In Theorem 1.4, let a = c. Then f (s) is a strictly increasing function for
d < b, or a strictly decreasing function for d > b. Thus Corollary 1.5 holds.
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References
[1] E.F. BECKENBACH
1961.
AND
R. BELLMAN, Inequalities, Springer Verlag,
[2] CHAO-PING CHEN AND FENG QI, An alternative proof of monotonicity
for the extended mean values, Aust. J. Math. Anal. Appl., 1(2) (2004), Art.
11. [ONLINE: http://ajmaa.org/].
[3] J.-CH. KUANG, Applied Inequalities, 2nd ed., Hunan Education Press,
Changsha, China, 1993. (Chinese)
[4] E.B. LEACH AND M.C. SHOLANDER, Extended mean values, Amer.
Math. Monthly, 85 (1978), 84–90.
[5] E.B. LEACH AND M.C. SHOLANDER, Multi-variable extended mean
values, J. Math. Anal. Appl., 104 (1984), 390–407.
Generalizations of the Ky Fan
Inequality
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[6] J.K. MERIKOWSKI, Extending means of two variables to several variables, J. Ineq. Pure. Appl. Math., 5(3) (2004), Art. 65. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=411].
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[7] D.S. MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, Berlin, 1970.
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[8] J. PEC̆ARIĆ AND V. S̆IMIĆ, The Stolarsky-Tobey mean in n variables,
Math. Inequal. Appl., 2 (1999), 325–341.
Close
[9] F. QI, Logarithmic convexity of extended mean values, Proc. Amer. Math.
Soc., 130(6) (2002), 1787–1796 (electronic).
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[10] K.B. STOLARSKY, Generalizations of the logarithmic mean, Math. Mag.,
48 (1975), 87–92.
[11] M.D. TOBEY, A two-parameter homogeneous mean value, Amer. Math.
Monthly, 87 (1980), 545–548. Proc. Amer. Math. Soc., 18 (1967), 9–14.
[12] ZH. WANG, J. CHEN AND X. LI, A generalization of the Ky Fan inequality, Univ. Beograd. Publ. Elektrotehn. Fak., 7 (1996), 9–17.
[13] CH.-P. CHEN AND F. QI, Monotonicity properties for generalized logarithmic means, Aust. J. Math. Anal. Appl., 1(2) (2004), Art. 2. [ONLINE:
http://ajmaa.org/].
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