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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 870950, 4 pages
doi:10.1155/2008/870950
Research Article
On Logarithmic Convexity for Ky-Fan Inequality
Matloob Anwar
1
and J. Pe
ˇ
cari
´
c
1, 2
1
Abdus Salam School of Mathematical Sciences, GC University, Lahore 54660, Pakistan
2
Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia
Correspondence should be addressed to Matloob Anwar, matloob
Received 19 November 2007; Accepted 14 February 2008
Recommended by Sever Dragomir
We give an improvement and a reversion of the well-known Ky-Fan inequality as well as some re-
lated results.
Copyright q 2008 M. Anwar and J. Pe
ˇ
cari
´
c. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
Let x
1
,x
2
, ,x
n
and p
1
,p
2
, ,p
n
be real numbers such that x
i
∈ 0, 1/2,p
i
> 0withP
n
n
i1
p
i
.LetG
n
and A
n
be the weighted geometric mean and arithmetic mean, respectively,
defined by G
n
n
i1
x
p
i
i
1/P
n
,andA
n
1/P
n
n
i1
p
i
x
i
x. In particular, consider the above-
mentioned means G
n
n
i1
1 − x
i
p
i
1/P
n
,andA
n
1/P
n
n
i1
p
i
1 − x
i
. Then the well-
known Ky-Fan inequality is
G
n
G
n
≤
A
n
A
n
. 1.1
It is well known that Ky-Fan inequality can be obtained from the Levinson inequality 1, see
also 2, page 71.
Theorem 1.1. Let f be a real-valued 3-convex function on 0, 2a,thenfor0 <x
i
<a, p
i
> 0,
1
P
n
n
i1
p
i
f
x
i
− f
1
P
n
n
i1
p
i
x
i
≤
1
P
n
n
i1
p
i
f
2a − x
i
− f
1
P
n
n
i1
p
i
2a − x
i
. 1.2
In 3, the second author proved the following result.
2 Journal of Inequalities and Applications
Theorem 1.2. Let f be a real-valued 3-convex function on 0, 2a and x
i
1 ≤ i ≤ n n points on
0, 2a,then
1
P
n
n
i1
p
i
f
x
i
− f
1
P
n
n
i1
p
i
x
i
≤
1
P
n
n
i1
p
i
f
a x
i
− f
1
P
n
n
i1
p
i
a x
i
. 1.3
In this paper, we will give an improvement and reversion of Ky-Fan inequality as well
as some related results.
2. Main results
Lemma 2.1. Define the function
ϕ
s
x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
s
ss − 1s − 2
,s
/
0, 1, 2,
1
2
log x, s 0,
−xlog x, s 1,
1
2
x
2
log x, s 2.
2.1
Then φ
s
xx
s−3
, that is, ϕ
s
x is 3-convex for x>0.
Theorem 2.2. Define the function
ξ
s
1
P
n
n
i1
p
i
ϕ
s
2a − x
i
− ϕ
s
x
i
− ϕ
s
2a − x ϕ
s
x2.2
for x
i
,p
i
as in 1.2.Then
1 for all s, t ∈ I ⊆ R,
ξ
s
ξ
t
≥ ξ
2
r
ξ
2
st/2
, 2.3
that is, ξ
s
is log convex in the Jensen sense;
2 ξ
s
is continuous on I ⊆ R,itisalsolog convex, that is, for r<s<t,
ξ
t−r
s
≤ ξ
t−s
r
ξ
s−r
t
2.4
with
ξ
0
1
2
ln
G
a
n
A
n
G
n
A
a
n
, 2.5
where G
a
n
n
i1
2a − x
i
p
i
1/P
n
, A
a
n
1/P
n
n
i1
p
i
2a − x
i
.
M. Anwar and J. Pe
ˇ
cari
´
c3
Proof. 1 Let us consider the function
fx, u, v, r,s, tfxu
2
ϕ
s
x2uvϕ
r
xv
2
ϕ
t
x, 2.6
where r s t/2, u, v, r, s, t are reals.
f
x
ux
s/2−3/2
vx
t/2−3/2
2
≥ 0 2.7
for x>0. This implies that f is 3-convex. Therefore, by 1.2,wehaveu
2
ξ
s
2uvξ
r
v
2
ξ
t
≥ 0, that
is,
ξ
s
ξ
t
≥ ξ
2
r
ξ
2
st/2
. 2.8
This follows that ξ
s
is log convex in the Jensen sense.
2 Note that ξ
s
is continuous at all points s 0,s 1, and s 2 since
ξ
0
lim
s→0
ξ
s
1
2
ln
G
a
n
A
n
G
n
A
a
n
,
ξ
1
lim
s→1
ξ
s
1
P
n
n
i1
p
i
x
i
ln x
i
−
2a − x
i
ln
2a − x
i
2a −
x ln2a − x − x ln x,
ξ
2
lim
s→2
ξ
s
1
2
1
P
n
n
i1
p
i
2a − x
i
2
ln
2a − x
i
− x
2
i
ln x
i
− 2a −
x
2
ln2a − xx
2
ln x
.
2.9
Since ξ
s
is a continuous and convex in Jensen sense, it is log convex. That is,
t − r ln ξ
s
≤ t − s ln ξ
r
s − r ln ξ
t
, 2.10
which completes the proof.
Corollary 2.3. For x
i
, p
i
as in 1.2,
1 < exp
2ξ
4
3
ξ
−3
4
≤
G
a
n
A
n
G
n
A
a
n
≤ exp
2ξ
3/4
−1
ξ
1/4
3
. 2.11
Proof. Setting s 0,r −1, and t 3inTheorem 1.2,wegetξ
4
0
≤ ξ
3
−1
ξ
3
or
ξ
0
≤ ξ
3/4
−1
ξ
1/4
3
. 2.12
Again setting s 3,r 0, and t 4inTheorem 1.2,wegetξ
4
3
≤ ξ
0
ξ
3
4
or
ξ
0
≥ ξ
4
3
ξ
−3
4
. 2.13
Combining both inequalities 2.12, 2.13,weget
ξ
4
3
ξ
−3
4
≤ ξ
0
≤ ξ
3/4
−1
ξ
1/4
3
. 2.14
4 Journal of Inequalities and Applications
Also we have ξ
s
positive for s>2; therefore, we have
0 <ξ
4
3
ξ
−3
4
≤ ξ
0
≤ ξ
3/4
−1
ξ
1/4
3
. 2.15
Applying exponentional function, we get
1 < exp
2ξ
4
3
ξ
−3
4
≤
G
a
n
A
n
G
n
A
a
n
≤ exp
2ξ
3/4
−1
ξ
1/4
3
. 2.16
Remark 2.4. In Corollary 2.3, putting 2a 1 we get an improvement of Ky-Fan inequality.
Theorem 2.5. Define the function
ρ
s
1
P
n
n
i1
p
i
ϕ
s
a x
i
− ϕ
s
x
i
− ϕ
s
a x ϕ
s
x, 2.17
for x
i
,p
i
,aas for Theorem 1.1.Then
1 for all s, t ∈ I ⊆ R,
ρ
s
ρ
t
≥ ρ
2
r
ρ
2
st/2
, 2.18
that is, ρ
s
is log convex in the Jensen sense;
2 ρ
s
is continuous on I ⊆ R,itisalsolog convex. That is for r<s<t,
ρ
t−r
s
≤ ρ
t−s
r
ρ
s−r
t
2.19
with
ρ
0
1
2
ln
G
n
A
n
G
n
A
n
, 2.20
where
G
n
n
i1
a x
i
p
i
1/P
n
,
A
n
1/P
n
n
i1
p
i
a x
i
.
Proof. The proof is similar to the proof of Theorem 2.2.
Remark 2.6. Let us note that similar results for difference of power means were recently ob-
tained by Simic in 4.
References
1 N. Levinson, “Generalization of an inequality of Ky-Fan,” Journal of Mathematical Analysis and Applica-
tions, vol. 8, no. 1, pp. 133–134, 1964.
2 J. Pe
ˇ
cari
´
c, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications,
vol. 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992.
3 J. Pe
ˇ
cari
´
c, “An inequality for 3-convex functions,” Journal of Mathematical Analysis and Applications,
vol. 90, no. 1, pp. 213–218, 1982.
4 S. Simic, “On logarithmic convexity for differences of power means,” Journal of Inequalities and Applica-
tions, vol. 2007, Article ID 37359, 8 pages, 2007.
